In Phase 10, if the person to my right uses their 'Skip card' to skip over me, can I still pick the top card off the discard pile?
No you cannot.
From the rules:
Skip Card – This card enables you to force someone to skip or lose a turn for that particular rotation/round. Once you discard or play it down, you can choose anyone to lose their turn. The Skip card may not be picked up from the Discard Pile, only from the Draw Pile. If it’s the first card to be turned up at the start of the game, the first player to start will lose a turn.