Over the years, I've gone from immediate skinning when I got home ranging to three days, if the temperatures are 50's or below. I field dress them immediately after they are retrieved, so no viscera are in the carcasses. From there on out, it is a matter of air temps and air flow. I put them out in the unheated mud room. If it is freezing, I get to it more quickly, just for the ease of the effort.
There was a time when folks would take week-long trips, spouse, kids and all, and get pheasants for the larder. They would set up a base camp and Mom and little kids would stay there, plucking, cooking and canning the pheasants harvested during the day. There were no limits, so this was protein shopping for families.
Then, too, some birds were hardy old survivors, the meat of which became more tender with "hanging," much like some deer hunters did/do. These days, I doubt we see many old birds, so letting the enzymes (?) work on the meat fiber is probably never required.
I skin all my birds and have done so for many years. There's just not enough palatable fat in the skin to go to the effort of plucking for me.
Basically, within reason, it is all up to you.
Have fun, be safe.