and a scour
ing pad on a good stainless steel or cast iron omelet
pan is the worst possible way of cleaning it. Well...maybe not the worst, but it will make an omelet stick.
Let it cool, use table salt and a towel (paper is okay) to clean it, then rinse and dry it. An omelet pan needs to be sealed with a layer of baked-on oil for a smooth finish, so coat the bottom of the pan with vegetable oil, heat over a medium temperature until the pan is really hot, but not smoking, then let it cool and put it away.
Of course, this doesn’t apply to non-stick pans. Cast iron or stainless steel pans really are better for making omelets. The consistency is quite different.