The more elements you add to an antenna array, the more directional or lobe structured it will become.  If you want to make your antenna more omni, remove one of the elements.  Remember, through, that you will reduce gain.  That's always the tradeoff.

If the antennas are combined at RF, then you've made an array and (as mentioned by 'yalamo'), you will tend to make a more complicated pattern (potentially with higher peak gain and more nulls).

One (expensive) trick is to detect the signals and combine them at baseband (after detection).  This is how some RF Field Probes work.  They can be very isotropic.

I would like to supply more information and get your input. I am working with an on/off keyed signal so there is no baseband and rf we are only dealing with one frequency, 13.56MHz.  I am not sure if array is the proper term when I am describing my idea.  (I get the idea in an array the elements are all oriented in the same direction.)  We currently use a ferrite core in a loop antenna in the receiver.  As I understand it the radiation/reception pattern for this antenna is a torroidal (doughnut) shape.  In my experiment I oriented three receivers each with an antenna in the x,y and z axis.  I combined the signals at the RSSI output of an amplifier and found I eliminated the nulls present in the current receiver.  So, I guess I am trying to find the cheapest way to combine three antennas for a more omnidirectional receiver.

In this context, an 'array' means any arrangement of antenna elements that are combined at RF.  They don't have to be aligned.  You're not the first to use multiple elements aimed in different directions to provide better coverage.

Combining antenna elements at RF tends to make the pattern more complicated (peaks and nulls).  You want an isotropic pattern which is as simple a pattern as can be imagined (a simple sphere) - but that can be very difficult to achieve in practice.  The isotropic antenna is often called 'the mythical isotropic antenna' (perhaps a bit harsh).

If you're using one ferrite core for your antenna, then it should only have two major nulls - one off each end (just like any old AM Broadcast Band receiver).  Therefore, you should only require (at most) one more ferrite core plus receiver to provide complete coverage (you shouldn't need three).

If you're dealing with an on/off keyed signal, then the baseband is basically DC.  That's the easiest possible baseband.  Depending on the on/off key rate, maybe you could have two of your ferrite cores arranged orthogonally, and then use a switched receiver back and forth between the two antennas.  This would allow your system to detect signals from any direction without having to pay for two complete receiver strips.  RF switching at 13MHz isn't rocket science.

Another option would be to abandon the linear ferrite core and investigate other options.  Active antennas work fairly well at HF frequencies.  They just need a short probe and a high impedance RF amplifier.  The probe might still have nulls.

Maybe a ferrite that is not linear?  I would guess that a curve ferrite might be somewhat null-free (a guess...).

What is your application (to the extent that you're permitted to talk about it)?  If I recall correctly, 13.56MHz is an 'Industrial' frequency and is often used for induction heating.  Are you trying to measure field strengths ?  If so, then just buy a suitable, calibrated probe...

The product as I inherited it is an 13.56MHz access control system.  It is used in the care of dementia patients who have a tendancy to wander off (potentially into harms way). The system is comprised basically of a small transmitter worn by the patient and a receiver at facility exits.  These components work together to allow free access by staff and visitors while restricting access (locking the door)by patients.  Both tx and rx use ferrite core antennas.  The combination of nulls in both the rx and tx lead to substantial descrepencies in detection range depending on orientation, angle of the tx with respect to the fixed rx.  The drive for ever smaller transmitters forces me to solve the problem at the receiver end.  As I stated before three recievers covering x,y and z axis perform quite well but is expensive.  I am intrigued by the post on rf switching and active antennas.  Any info you can provide on these topics would be appreciated.  Thanks to all participating in this topic.

RF switching would simply mean designing the receiver to alternate between two or three antennas and take readings with each one.  Given the system description, it would seem that the directionality of the receiver is something that could be dealt with (aim it at the doorway).

Active antennas are simply a short probe fed into a very high impedance input such as a MOSFET gate.  Very common as an accessory for shortwave listening.  Even Sony sold one at one time (the AN-1).

Given the overall system, I'd simply trash the whole system, lock the doors with magnetic locks, and hand out off-the-shelf proximity fobs to the staff.  The whole thing is standard "access control" and is cheap.