Like Bruce, keeping a soap uncovered will change it's scent, sometimes (for me) for the better. Other than deliberately leaving a soap open to reduce it's scent I use it and immediately close it. I have over 20 soaps in "use" right now so they can't be kept uncovered. Some soaps will change over time and not for the better. Cella is one that will change and darken in a few years; others take longer. Hard soaps are much longer lasting; I'm using one right now from the '70s that is tan and subdued in scent, but excellent. Soaps in tubes (cream) will sometimes age badly, I know that Ingrams once exposed to heat will NOT turn back into a cream but will stay like a thick liquid.
You asked about maintenance... If it's in a tub any unused proto lather just remains in the tub when I screw on the lid. After all, it's simply soap just waiting for the next use. Most of what I worked up comes out with the brush.
In the end it's up to you how you do it, but generally I'm not in favor of exposing soap to the air any longer than required to work up a proto lather. Lots of folks will disagree with me, and if they don't I'll be very surprised. Who's right? Everyone. As with many things in shaving, YMMV. Just do what you think is best and you'll be correct.
I hope that helps.
Brian. Lover of SE razors.