Skip card points count against you?
No, if you played/discarded the skip card on your turn then it is no longer in your hand; only cards that are in your hand count against you.
The rules say

Players total the cards left in their hands.

and

The winner of the hand scores zero. All remaining players score points against them, for cards still in their hands, as follow:

Note that there's really no difference between discarding a skip card on your turn and discarding any other card, in terms of how it affects you. The card is only placed in front of a specific player to remind them that their turn is getting skipped. Older versions of the rules don't say to do this. It doesn't matter if the card is physically in the discard pile or in front of a player; it has already been played/discarded by you.