Determinant of a matrix with consecutive integers

The following operations are transvections and hence do not change the determinant :
C3 becomes C3-C2 then C2 becomes C2-C1.
In the new matrix, C2=C3 (colums made up of 1) so the determinant is zero when n>=3.

For n=1, the determinant is 1.
For n=2, it is -2 (classic formula for a size 2 matrix).