The hyperbolicity cones associated with the elementary symmetric polynomials provide an intriguing family of non-polyhedral relaxations of the non-negative orthant which preserve its low-dimensional faces and successively discard higher dimensional facial structure. We show by an explicit construction that this family of convex cones (as well as their analogues for symmetric matrices) have polynomial-sized representations as projections of slices of the PSD cone. This, for example, allows us to solve the associated linear cone program using semidefinite programming.