Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish.

I remember hearing that one really does not need to check all possible minors in order to conclude that all of them would vanish.

If such a result is true, how many minors will do the job and which ones ?

I am wondering if it is even possible to calculate the value of all minors based on the value of a nicely chosen "generating subset" ?