If $C_1$ and $C_2$ are circuits in a matroid $M$ with $e_1$ in $C_1-C_2$ and $e$ in $C_1\cap C_2$, then $M$ has a circuit $C_3$ such that $e\in C_3\subseteq (C_1\cup C_2)-e$. This strong circuit elimination axiom is inherently asymmetric. A matroid $M$ has the symmetric strong circuit elimination property (SSCE) if, when the above conditions hold and $e_2\in C_2-C_1$, there is a circuit $C_3'$ with $\{e_1,e_2\}\subseteq C_3'\subseteq (C_1\cup C_2)-e$. We prove that a connected matroid has this property if and only if it has no two skew circuits. We also characterize such matroids in terms of forbidden series minors, and we give a new matroid axiom system that is built around a modification of SSCE.