How many symmetric and non-symmetric $n\times n$ matrices with $0/1$ entries are there such that every row is distinct and every column is distinct? (I am looking for a proof as well).

If only every row (or column) is distinct is needed, the answer is easy.

As suggested in comments, https://oeis.org/A088310 provides the numbers without symmetric restriction. I do not see a proof or link for proof there.

What about symmetric case?