Assume that we have the polynomial ring $k[a_{ij}]$ where $a_{ij}$ are the symbolic entries of a $m\times n$ matrix where $m\leq n$. What is the codimension of the variety that is the zero set of the determinants of $m\times m$ submatrices?
My thoughts : I want the $m$ rows of the matrix to be linearly dependent. Hence I am free to fill the first $m-1$ rows however I want and then for the last row I need to choose $m-1$ coefficients so that the last row is the sum of the first $m-1$ multiplied with these coefficients. Hence the dimension should be $(m-1)\times (n+1)$ which gives me the codimension $n-m+1$. When I apply it to the well-known codimension $2$ case ($m=n-1$) I get the correct result.
Even more generally take your matrix and look at the subvariety defined by the vanishing of all $r\times r$ submatrices, $r\leq m$. Then the codimension is $(m-r+1)\cdot (n-r+1)$. This is a theorem by Eagon and Northcott.