I'm trying to understand how to decompose a high $N$-dimensional Hilbert space $V$ thinking it as a subspace of the tensor product space $V_1 \otimes V_2$ of two smaller Hilbert spaces $V_1$ and $V_2$, respectively of dimension $n_1$ and $n_2$.
Let $\{{|o_i\rangle}\}_{i=1}^N$ be a complete basis that spans the $V$ space and let $\{{|p_i\rangle}\}_{i=1}^{n_1}$, $\{{|m_i\rangle}\}_{i=1}^{n_2}$ be two complete bases that span respectively the $V_1$ and the $V_2$ spaces. As far as I know, I can always build a basis for $V_1 \otimes V_2$ from the bases of the lower dimensional spaces $V_1$ and $V_2$ as $\{p_\alpha \otimes m_\beta \,|\, p_\alpha \in \{{|p_i\rangle}\}_{i}, m_\beta \in \{{|m_j\rangle}\}_{j} \}$.
So then I can write any $|\phi\rangle \in V$ as: $$|\phi\rangle = \sum_{i=1}^N \phi_i |o_i\rangle=\sum_{l=1}^{n_1} \sum_{k=1}^{n_2} \Phi_{lk} \, |p_l\rangle |m_k\rangle \;\;\;\;\;\;\;\;\;\;\;\;\; (\Box)$$
Question: Starting from $|\phi\rangle \in V$ and its representation in the $\{{|o_i\rangle}\}_{i=1}^N$ basis, how can I build such mapping? I tried to decompose every side of $(\Box)$ into components and match them one by one, but it is a very cumbersome approach: I was wondering if there is a better and more general alternative which allows me to work only in the $V$ space.
In fact notice that it's not always possible to find a factorization of $N$ as the product $n_1 \cdot n_2$, so then in general one has $N\leq n_1 \cdot n_2$ and $V \subseteq V_1 \otimes V_2$.
Edit: The fact is that I would like to decompose the space $V$ without adding further dimensions, and therefore without extending it to the tensor product space $V_1 \otimes V_2$. I need a sort of tensor space truncation procedure, but probably it doesn't make any sense and is not mathematically rigorous.