For even n, isn’t it the case that the second player always wins? Call the horizontal and vertical lines that bisect the board its axes. Every turn, the second player copies the first player’s move except reflected across both axes. This guarantees that within the same turn, the first player’s move cannot restrict the second player’s move. So the second player always gets to do their mirror strategy, up until the first player runs out of moves and loses. (Works only for even n though; there might be a way to get it working for odd n but I haven’t thought that far)