Let $e_i$ denote a column-vector of length $n$ whose entries are all zero except for the $i$-th entry that is 1. Now consider the set of $n\times n$ matrices given by $\mathcal{M}_n=\left\lbrace \left(e_i-e_j\right)\left(e_i-e_j\right)^\mathrm{T}\mid 1\leq j
My question is that can we obtain all $n\times n$ real symmetric positive-semidefinite matrices as conic combinations of matrices in $\mathcal{M}_n$?