Indeed, the Ski-Rook is a major piece. You can practice checkmating with it here.
If it only has a D move in one of the dimensions it is more problematic, whether it skips the first square in the other dimension or not. Because in that case it is bound to even or odd ranks or files. On an even-size board there than always is one edge it cannot attack, and if the bare King can reach there it has a fortress draw. If not, it is a forced win.
Indeed, the Ski-Rook is a major piece. You can practice checkmating with it here.
If it only has a D move in one of the dimensions it is more problematic, whether it skips the first square in the other dimension or not. Because in that case it is bound to even or odd ranks or files. On an even-size board there than always is one edge it cannot attack, and if the bare King can reach there it has a fortress draw. If not, it is a forced win.
But the Ski-Rook can reach the entire board.