Let $E \to G$ be the tautological bundle of $n$-planes, and let $p : Fl(E) \to G$ be the bundle of complete flags in $E$. Tautologically, the pullback $p^\star E \to Fl(E)$ is the universal vector bundle of rank $n$ with a choice of complete flag.
On the other hand, since short exact sequences of vector bundles over paracompact spaces always split, the isomorphism type of a complete flag can be recovered from its line bundle quotients. Therefore, the $n$-fold product of projective spaces $(\mathbb P^\infty)^n$ also classifies vector bundles of rank $n$ with a choice of complete flag.
How can I construct an explicit homotopy equivalence between $Fl(E)$ and $(\mathbb P^\infty)^n$?