  Using quantum skew-Howe duality, we study the category
$\operatorname{Rep}(\mathfrak{gl}(m|n))$ of tensor products of exterior powers
of the standard representation of $U_q(\mathfrak{gl}(m|n))$, and prove that it
is equivalent to a category of ladder diagrams modulo one extra family of
relations. We then construct a categorification of
$\operatorname{Rep}(\mathfrak{gl}(m|n))$ using the theory of foams. In the case
of $n=0$, we show that we can recover $\mathfrak{sl}(m)$ foams introduced by
Queffelec and Rose to define Khovanov-Rozansky $\mathfrak{sl}(m)$ link
homology. We also define a categorification of the monoidal category of
symmetric powers of the standard representation of $U_q(\mathfrak{gl}(n))$,
since this category can be identified with
$\operatorname{Rep}(\mathfrak{gl}(0|n))$. The relations on our foams are
non-local, since the number of dots that can appear on a facet depends on the
position of the dot in the foam, rather than just on its colouring. This may be
related to Gilmore's non-local relations in Heegaard Floer knot homology.
