I suppose this would be pretty difficult with "live" recorded sound. Stereo microphone arrays are a complex business already, AFAIK there's no audio equivalent of the 360 degree cameras they use for recording VR. Microphones simply cannot be made as directional as speakers.

However with a multichannel surround sound recording, panning/fading the tracks dynamically according to head movements is trivial, all games do this. So I guess you would want a file format that placed all the channels in 3D space over time. That's the kind of thing the spat software I linked in the above post does, except it's also doing cutting edge acoustic modelling and so forth. It needs a hefty computer.