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?

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

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?

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?

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?

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