If I want to write an $m\times n$ $0/1$ matrix with only rows **or** columns distinct, I could just pick $m$ or $n$ distinct natural numbers effectively writing them down as rows or columns in base $2$.

Is there a canonical way to write down an $m\times n$ $0/1$ matrix of rank $r$ such that every row is distinct **and** every column is distinct? Case $m=n$ is most interesting.

If not, what are some tricks and strategies to obtain such a matrix of rank $r$?