When a Gate is closed you put any monsters with the Gate symbol back into the monster mix, do you include monsters in the Sky and the Outskirts.

The answer is yes, the rules are very clear on this question.

"When a gate is closed, all monsters in Arkham, the Sky and the Outskirts that have the same dimensional symbol as the closed or sealed gate are removed from the board and returned to the cup." (pg 18 of Arkham Horror rulebook).