Abstract: We introduce a new presentation for the Chow ring of a matroid in terms of variables whose action can be interpreted combinatorially in terms of matroid quotients. Applications of this property include a new proof of Poincar\'e duality for the Chow ring, a combinatorial interpretation for a monomial basis of Feichtner and Yuzvinsky, a formula for the volume polynomial of the Chow ring, and a new proof of the Heron-Rota-Welsh conjecture.