Have you read this link?
plasma can be created by heating a gas or subjecting it to a strong electromagnetic field, applied with a laser or microwave generator at temperatures above 5000 Celsius. This decreases or increases the number of electrons, creating positive or negative charged particles called ions, and is accompanied by the dissociation of molecular bonds, if present.
So a plasma is gas of electrons and the ions left behind when exiting to create the plasma. The atoms do not lose their identity by ionizing and will recombine into neutral atoms when cooled.
if plasma is just a pool of unbound electrons, protons and neutrons?
Given enough energy , maybe the strong bonds of nucleons in the nucleus could be broken and temporarily a gas at high temperatures will exist as you think. The minute the neutrons are free they will decay into a proton and electron and an electron antineutrino, so the plasma will consist of protons and electrons in a few lifetimes (881.5 seconds). When the plasma cooled it would recombine into hydrogen.
There exists the proposal of a quark gluon plasma, which is searched for in experiments at the LHC and is important for cosmological models, but that is another story.