In Phase 10, can a Skip card be played in the first round of a hand?
There is no such restriction. From the rules:


  The player ends the turn by discarding one card of choice onto the discard pile.


And 


  SKIP CARDS: When discarded, a "Skip " card causes another player to lose their next turn. The player who discards the "Skip " card chooses the player who loses their turn. When a player draws a "Skip " card, the player may discard it immediately or save it for a later turn. A "Skip " card may never be used in making Phase 8, or any other Phase. A "Skip " card may never be picked up from the discard pile.