Geometry of some determinantal arrangements

Let $D_{m,n}\subset\C^{m\times n}$ be the union of the zero loci of the maximal minors of a generic $m\times n$ matrix, and let $U_{m,n}$ be its complement. We show that $U_{m,n}\cong\GL_m(\C)\times\Gr(m,n)^\circ$, where $\Gr(m,n)^\circ$ is the open stratum of the Grassmannian on which all Plücker c