Storing temperature for cooked omelette
You could keep it frozen for longer than a week, but by the time you reheated it, it would be like rubber.
I wouldn't attempt more than a couple or three days in the fridge.

The only way I can think to keep anything omelette-like would be like a Spanish tortilla - maybe it's the potato &amp; onion that changes the texture, but that is fine after being cooled then re-heated.