It's not an easy thing to do. The first thing we usually do is separate the system into the eigenmodes and TM/TE modes. That is, for many problems, we can separate the electromagnetic solutions into two sets. One set is TE_z, characterized by the H_z field, and the other is TM_z, characterized by the E_z field. This way, we only need to solve for the E_z and H_z fields to fully describe the system. The rest of the fields are taken as the cross products of these fields and of course you can orient the fields as needed by your problem.
The next problem is that the solutions to a resonant cavity, among other problem geometries, can be described as the superposition of eigenmodes. So if you were to just put in a random excitation into the cavity, it could be described, once the transients died out, as a liner superposition of various eigenmodes.
So how would we visualize these solutions? Usually we map out the magnitude and phase of the field components. These two values will encompass the full amount of information. You could also do the real and imaginary parts too. Either way, you could do field plots over 2D cross-sections of the eigenmodes for the H_z and E_z fields. That would be one way to visualize a large amount of information for the cavity solution. You will typically see these kinds of visualizations for waveguides, especially because a waveguide is typically solved assuming that the guided direction is invariant (infinite). Thus, for a waveguide a 2D cross-section can fully show the fields but with a 3D cavity that is not true.
As to a 3D representation, no, you cannot do a true 3D representation very well. The problem is that the field is spatially dependent but it's values are a fourth and fifth dimension. We are interested in the field vector at all points so there is more information than can be shown in 3D. The best you could do is a flow plot in 3D space, this would have arrows that represent the direction and magnitude of the field vectors but since the fields are complex you would still lose the phase information in some way or another. But typically, signed magnitude would provide a good picture so you could do it that way, Matlab has ways of doing this but I have always found them to be cumbersome.