What game mechanic can be used for evenly selecting other players but not yourself?
If it is acceptable that the order of the players turns is different every round, you could have a deck with a card for each player, then draw the top card and let that player have the next card as target, and then the next player takes the third card as target.
Finally the last player uses the first one as target. This way each player is the target exactly once in every round.