The principal minor map takes an n by n matrix to the vector of its 2^n principal minors. For complex symmetric matrices, equations cutting out the image of this map were described by Oeding in 2011. In this talk, I will describe a generalization of this result to matrices over arbitrary fields and some spaces of non-symmetric matrices. This characterization comes from a characterization of determinantal representability of multiaffine polynomials. This is based on joint work with Abeer Al-Ahmadieh.