  The semiclassical $\hbar$-expansion of the one-particle density matrix for a
two-dimensional Fermi gas is calculated within the Wigner transform method of
Grammaticos and Voros, originally developed in the context of nuclear physics.
The method of Grammaticos and Voros has the virture of preserving both the
Hermiticity and idempotency of the density matrix to all orders in the
$\hbar$-expansion. As a topical application, we use our semiclassical expansion
to go beyond the local-density approximation for the construction of the total
dipole-dipole interaction energy functional of a two-dimensional,
spin-polarized dipolar Fermi gas. We find a {\em finite}, second-order gradient
correction to the Hartree-Fock energy, which takes the form $\varepsilon
(\nabla \rho)^2/\sqrt{\rho}$, with $\varepsilon$ being small ($|\varepsilon|
\ll1$) and negative. We test the quality of the corrected energy by comparing
it with the exact results available for harmonic confinement. Even for small
particle numbers, the gradient correction to the dipole-dipole energy provides
a significant improvement over the local-density approximation.
