When playing phase 10 with 2 players. Player one goes out and discards the skip, does player two get to play the run in their hands for the phase they’re on or do they have to take the points for not laying down their hand their last turn before they were skipped?
You seem to have a few things mixed up about the rules. The answer to the question is “no”, but it has nothing to do with the skip card. The round ends when a player goes out, either by discarding their last card or by hitting on an existing phase to get rid of their last card. When player 1 went out, the round was over, and player 2 does not get a turn. It is irrelevant if the last card discarded was a Skip card or not.
In addition to that, player 2 should not have been holding cards they could have played. If they had what they needed for their phase in their hand, then they would have played it on their previous turn. If they don’t have the required cards for their phase, then they wouldn’t be allowed to lay down any cards even if they did get another turn. You have to complete your phase before you can lay down any other cards.