[Math] On full rank matrix properties

Question

$(1)$ What is number of rank $n$ $m\times n$ $0-1$ matrices when $m=n$ and $m>n$? Is there solutions closed forms?

$(2)$ Over $\Bbb F_2$, does full rank of an $n\times n$ matrix imply determinant is non-zero and vice versa?

Answer 1

For $m=n$ you might look at OEIS sequence A055165 and references there, in particular the Živković article. Bottom line: there seems to be no known closed form, but there are conjectured asymptotics.

EDIT: For question (2): Over any field, full rank of an $n \times n$ matrix is equivalent to the determinant (evaluated in that field) being nonzero.