A pseudo-3D modeling approach, based on a particle-in-cell (PIC)-Monte Carlo collisions algorithm, has been developed for the study of large- and short-scale organization of the plasma in a planar magnetron. This extension of conventional PIC modeling permits the observation of spontaneous organization of the magnetron plasma, under the influence of crossed electric and magnetic fields, into the well-known, large-scale regions of enhanced ionization and density known as spokes. The nature of complex three-dimensional electron trajectories around such structures, and non-uniform ionization within them, is revealed. This modeling provides direct numerical evidence for the existence of high-amplitude internal spoke electric fields, proposed in earlier works. A 3D phenomenological model,consistent with numerical results, is proposed. Electron density fluctuations in the megahertz range, with characteristics similar to the electron cyclotron drift instability experimentally identified in a recent Letter, are also found.