Yannakakis established a link between extended formulations of a given polytope and nonnegative factorizations of its slack matrix. In particular, the size of the smallest extended formulation is given by the nonnegative rank of that slack matrix. This was recently generalized by Gouveia et al. and Fiorini at al., where cone lifts of convex bodies are linked to factorization of slack operators. In this work, we investigate the use of this framework to answer the question of the second-order cone representability of the positive semidefinite cone. It is well-known that second-order cones can be represented with positive semidefinite cones: more specifically second-order cones are specific slices (i.e. intersection with a linear subspace) of the positive semidefinite cone. Not too much is known about the converse relationship. For example, we are not aware of any second-order representation of the 3x3 positive semidefinite cone (although some specific slices of that cone admit such a representation). Note that two distinct questions can be considered: representation using slices of a product of second-order cones, or representation as a projection (linear map) of such a slice (which allows more freedom). We will provide a negative answer to the first question, and discuss the second.