On my main tank, YES Skimmer is a must (to me at least)
I have a few 10G nano's in the garage though that have NO skimmer, NO filters... Just a heater, lights, and a few powerheads.
I haven't even changed the water in one of them for almost a year and it is going strong. Zoo's, Mushrooms, and a Green BTA.
All are doing great. I also haven't fed the tank in about 6 months.
I think the key here is the no feeding.
No nutrients in = no nutrients to take out.
Almost like the freshwater system in the sealed 2L bottle the kids do in 3rd grade.
Actually, speaking of that, you gave me an idea.
I have a 5G glass carboy (like a bottle you put on the water cooler, but glass and smooth (no ridges). I may try a system in one of those for kicks. Use natural sunlight, some snails, Calupera, and maybe some mushrooms and a Damsel.
Getting the live rock in there might be tough though.
Sorry... I digress...
So Yes, you could go without a skimmer, but you will have to be careful about how much you feed.