[seqfan] Matrices of permuted [1,2,..n^2] with zero determinant

[Richard J. Mathar]

The number of nXn matrices with entries forming a permutation of [n^2]
is (n!)^2. Determinants are characterized in http://oeis.org/A085000
and OEIS entries referenced there.
The number of these matrices with determinant equal to zero is 0 (n=1),
0 (n=2), 2736 (n=3) and by rough estimation of sampling a small subset
of the matrices of the order 1.5*10^9 for n=4.

Is there a better estimate
for the count of matrices in this class with determinant zero?

[Background: This yields a maximum of how many matrices with distinct entries
are convertible into each other (that is, convertible into the reference
matrix that has the 1,2,...,n^2 in order) by column and/or row 
exchanges/transpositions, because leave the absolute value of
the determinant invariant. (The reference matrix has determinant
1 (n=1), -2 (n=2) and apparently 0 otherwise.)]