Abstract: This paper is devoted to the classification of $4$-dimensional Riemannian spin manifolds carrying skew Killing spinors. A skew Killing spinor $\psi$ is a spinor that satisfies the equation $\nabla_X\psi=AX\cdot\psi$ with a skew-symmetric endomorphism $A$. We consider the degenerate case, where the rank of $A$ is at most two everywhere and the non-degenerate case, where the rank of $A$ is four everywhere. We prove that in the degenerate case the manifold is locally isometric to the Riemannian product $\mathbb{R}\times N$ with $N$ having a skew Killing spinor and we explain under which conditions on the spinor the special case of a local isometry to $\mathbb{S}^2\times\mathbb{R}^2$ occurs. In the non-degenerate case, the existence of skew Killing spinors is related to doubly warped products whose defining data we will describe.