As we know, the Projective space P^n represent the functor sending X to the set of line bundles L on X together with a surjection from the trivial vector bundle to L.

My question is, what functor does the Grassmannian Gr(d,n) represent? Sending X to the set of rank-$n$ vector bundles together with a subbundle of rank $d$?

More generally, what functor do flag varieties represent?