Yes. You have that type of notation in correspondence chess.
It goes something like this. Every square have a 2-digit number. The first digit is the column and the second digit is the row. That means that square e1 = 51. And for example b6 = 26.
When you write a move, you are writing the square it is starting from and the square it is going to. Say, your first move is e2-e4 in this notation is 5254. The castling, you are writing the king moves 5171.
The only time you put on the 5:th digit is when you are promoting a pawn.