Cones of positive semidefinite matrices generated by matrices of rank $1$

Let $S_n$ be the space of real $n \times n$ symmetric matrices and let $S_n^+$ be the convex cone of positive semidefinite matrices in $S_n$. The extremal rays of this cone correspond to the positive semidefinite matrices of rank $1$.

My problem is that I have a subcone $C$ of $S_n^+$ which is given by (finitely many) inequalities, which are not all linear (quadratic for example). In $C$ I have another subcone $C'$ given by generators which correspond to rank $1$ matrices. Since I would like to compare $C$ and $C'$, it would be nice to have inequalities defining $C'$. So my question is if there are known methods to determine such inequalities.

Any closed convex set in a locally convex topological vector space over $\mathbb R$ is the intersection of some family of half-spaces. In particular, any convex cone in ${\mathbb R}^{n \times n}$ is defined by a collection of linear inequalities (not necessarily equations). Moreover, since the space is separable, a countable collection will do. I don't know what else you would expect to say about it.