Dominoes on a chessboard

Is it always possible to remove N squares from an NxN chessboard in such a 
way that the remaining squares can be covered by N*(N-1)/2 dominoes in a 
unique way?

For example, for N=8, there is only one way of covering the following
board, 
where '-' represents a removed square, and '+' represents a remaining 
square:

-+++++++
+-+-++++
++++-+++
++++-+++
+++++-+-
++++++++
++++++++
++++++-+

[This would look better using a non-pro****tional font.]
--
Colin