Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the form $$ g = \nabla \cdot S + \nabla h, $$ where $h$ is harmonic and $S$ takes values in the set of skew-symmetric matrices. Does someone know of a good reference for this result?