What can I do with the fat skimmed from chicken stock?
Yes, it is really chicken fat rendered during the stock making process.

Called schmaltz in Yiddish, it is an ingredient in its own right.  For example, you can use it to fry foods, or instead of butter in creating a roux, when you would like the chickeny flavor it provides.   It is a key ingredient in matzo balls, and similarly, makes spectacularly good dumplings of various sorts. You can refrigerate it for several months, tightly covered.  

It is not necessary to return it to the stock, although you may use it as an additional ingredient in the dishes you make with the stock.

Stocks are normally defatted anyway, so you do not need to return it to your stock.