In creating the plasma in the first place, in the discharge (current of electrons), the electrons strike the atomic electrons knocking them out of the atom, if I may put it so crudely. However, the free electrons will slow by collision and will recombine with an ion.
In order to maintain a plasma, one usually needs to maintain the discharge, or heat the plasma to the point where the collisions maintain a certain level of ionization. One the achieves a balance between ionization and recombination, which is a function of the discharge current or plasma temperature.
As for using a static electric field, the electrons would drift to the positive electrode and ions to the negative electrode. The ions would then neutralize on the negative electrode.
Now theoretically, one could put the air in a chamber, and ionize it in the presence of a static electric field. Presambly one would end up with + and - ions, e.g. O2+ and O2- or N2+ and N2-, however I am not sure about the stability of diatomic ions. Perhaps there is a some monatomic ions as well, and I believe triatomic O3, aka ozone, is possible. However, the presence of + and - ions is not the same as + ions and free electrons, which is the conventional meaning of a plasma.