Most differentiable causal discovery approaches constrain or regularize an optimization problem using a continuous relaxation of the acyclicity property. The cost of computing the relaxation is cubic on the number of nodes and thus affects the scalability of such techniques. In this work, we introduce COSMO, the first quadratic and constraint-free continuous optimization scheme. COSMO represents a directed acyclic graph as a priority vector on the nodes and an adjacency matrix. We prove that the priority vector represents a differentiable approximation of the acyclic orientation of the graph, and we demonstrate the existence of an upper bound on the orientation acyclicity. In addition to being asymptotically faster, our empirical analysis highlights how COSMO performs comparably to constrained methods for graph discovery.