There is a thing called Majorana representation of the symmetric states, somehow related to your question.

For $\dim H = 2$ and $\psi$ living in a symmetric subspace of $H^{\otimes n}$, we have

$$\psi = \sum_{\hbox{perm}} \phi_{P(1)}\phi_{P(2)}\cdots \phi_{P(n)},$$ where $\phi_i \in H$. The representation is unambiguous (leave alone constant factors and the permutation of the indices). Explicit decomposition employs finding roots of a polynomial.

The bad thing is I do not know if there is a generalization for $\dim H > 2$ (a naive one does not work).

Schmidt decomposition is very relevant to quantum entanglement. Nevertheless, there is no straightforward generalization for $n>2$ (or in your words - for multilinear forms). See e.g.:

• A. Acin et al, Generalized Schmidt decomposition and classification of three-quantum-bit states, arXiv:quant-ph/0003050

There is a thing called Majorana representation of the symmetric states, somehow related to your question.

For $\dim H = 2$ and $\psi$ living in a symmetric subspace of $H^{\otimes n}$, we have

$$\psi = \sum_{\hbox{perm}} \phi_{P(1)}\otimes\phi_{P(2)}\otimes\cdots\otimes \phi_{P(n)},$$ where $\phi_i \in H$. The representation is unambiguous (leave alone constant factors and the permutation of the indices). Explicit decomposition employs finding roots of a polynomial.

The bad thing is I do not know if there is a generalization for $\dim H > 2$ (a naive one does not work).

Schmidt decomposition is very relevant to quantum entanglement. Nevertheless, there is no straightforward generalization for $n>2$ (or in your words - for multilinear forms). See e.g.:

• A. Acin et al, Generalized Schmidt decomposition and classification of three-quantum-bit states, arXiv:quant-ph/0003050