As we know, the projective space $\mathbb{P}^n$ represents 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 sub-bundle of rank $d$?

More generally, what functors do flag varieties represent?