How to decompose a vector into a sum of subvector with kronecker product？

Question

Define a random $15 \times 15$ square matrix $\bf{A}$ which is full rank. If ${\bf{w}} = vec({\bf{A}})$, I want to know how to decompose $\bf{w}$ by subvector via kronecker product, i.e., ${\bf{w}} = \sum\limits_i {{{\bf{g}}_i}} \otimes {{\bf{g}}_i}$, where ${{\bf{g}}_i}$ is a vector with 15 elements, ${{\bf{w}}}$ is a vector with 225 elements and $ \otimes $ is kronecker product. The cardinality of set $\{ {{\bf{g}}_i}\}$ is undetermined.