waste -> dissolved organic matter -> (eventually) nitrates
A skimmer helps to remove dissolved organic matter, before it can become nitrate.
While a skimmer dosn't remove nitrates directly, they do stop nitrates from being formed. So a skimmer (in the long term) would help to greatly reduce your nitrates, and in conjunction with water changes or vegetation, can help to keep your nitrate levels down, or even undetectable.
In a tank with no skimmer (and fish) I would expect to always see either a) detectable nitrates or b) large algae blooms.