Decomposing matrix as tensor product

Question

I have a matrix $F\in \mathbb{R}^{n\times n^2}$ for some integer $n>1$. I know in some cases, I can decompose $F$ as $$F = \mathbf{v}^\dagger\otimes \mathcal{F}$$ where $\mathbf{v}\in \mathbb{R}^n$ and $\mathcal{F}\in \mathbb{R}^{n\times n}$. In general, though, it's not possible to do this, and I have some counterexamples showing this is not possible. However, is it possible to decompose $F$ as: $$F = \sum_{j = 1}^r \mathbf{v}_r^\dagger \otimes \mathcal{F}_r$$ Or is this also not possible? Is there a relevant theorem or property I need to know? If this decomposition is possible, is it possible to bound $r$ in terms of $n$?