  We develop an operator commutant version of the First Fundamental Theorem of
invariant theory for the general linear quantum group $U_q(\mathfrak{gl}_n)$ by
using a double centralizer property inside a quantized Clifford algebra. In
particular, we show that $U_q(\mathfrak{gl}_m)$ generates the centralizer of
the $U_q(\mathfrak{gl}_n)$-action on the tensor product of braided exterior
algebras $\bigwedge_q(\mathbb{C}^n)^{\otimes m}$. We obtain a multiplicity-free
decomposition of the $U_q(\mathfrak{gl}_n) \otimes U_q(\mathfrak{gl}_m)$-module
$\bigwedge_q(\mathbb{C}^n)^{\otimes m} \cong \bigwedge_q(\mathbb{C}^{nm})$ by
computing explicit joint highest weight vectors. We find that the irreducible
modules in this decomposition are parametrized by the same dominant weights as
in the classical case of the well-known skew $GL_n \times GL_m$-duality.
Clifford algebras are an essential feature of our work: they provide a unifying
framework for classical and quantized skew Howe duality results that can be
extended to include orthogonal algebras of types $\mathbf{BD}$.
