Winner of impartial King placing game

Question

The game is played on an $n\times n$ board. Two players take turns placing kings, such that no two kings attack each other. The last player to move wins. If $n$ is odd, I think the game is a win for the first player, who starts by placing a king in the center. Then a pairing strategy works, where for every move of the second player, the first player rotates it 180 degrees around the center and moves there. However, I am not sure how to approach even $n$. For small cases $n=2$ is a win for the first player, $n=4$ is a loss, and $n=6$ is a win, so I guessed that the first player wins iff $\frac{n}{2}$ is odd, but I don't know how to prove it.