Sample $n^2$ integers $a_{11},\dots,a_{nn}$ in $[-d,d]\not\subseteq[0,0]$ uniformly.

What is the probability that the resulting matrix $[a_{ij}]$ has rank $r$?

Is there a nice parametrization of such matrices that helps us generate such a rank $r$ matrix quickly deterministically?

In general what is a good strategy?