Both are used. It is hard to explain the difference.
We use "the sky" when we are thinking of the dome itself, which is always there and is always the same "thing."
We use "the skies" when we are thinking of more than one pattern of stars or clouds. We use "the skies" when we are thinking of taking more than one look at the sky, at more than one time.
Here are some examples of use:
1) When I vacationed in Colorado, I was aware of the "big sky" out West. When I vacationed in Wyoming I again noticed that the sky seemed bigger than it does at home near Boston. I saw western skies on two vacations, and I understood why Katharine Lee Bates wrote "O beautiful for spacious skies..." after visiting Pike's Peak.
2) Lick Observatory was built on a mountaintop and enjoys clear skies almost every night. Yerkes Observatory was built at the side of a lake at sea level. It has been less productive than Lick because the skies are not as clear or as dark.