  We reconstruct the quantum enveloping superalgebra ${\bf
U}(\mathfrak{gl}_{m|n})$ over $\mathbb Q(v)$ via (finite dimensional) quantum
Schur superalgebras. In particular, we obtain a new basis containing the
standard generators of ${\bf U}(\mathfrak{gl}_{m|n})$ and explicit
multiplication formulas between the generators and an arbitrary basis element.
