We readdress the problem of nonparametric statistical testing of random utility models proposed in Kitamura and Stoye (2018). Although their test is elegant, it is subject to computational constraints which leaves execution of the test infeasible in many applications. We note that much of the computational burden in Kitamura and Stoye's test is due to their test defining a polyhedral cone through its vertices rather than its faces. We propose an alternative but equivalent hypothesis test for random utility models. This test relies on a series of equality and inequality constraints which defines the faces of the corresponding polyhedral cone. Building on our testing procedure, we develop a novel axiomatization of the random utility model.