Let $H$ be a (the) real separable Hilbert space. The Hilbert--Schmidt theorem says that a compact self-adjoint operator $A$ has an eigenfunction expansion. Instead of operator, we can think of a symmetric bilinear form and write $$ A = \sum_{k\ge 1} \lambda_k \varphi_k\otimes \varphi_k \tag{1} $$

My question is:

Are there any multilinear analogues of (1)? Which $n$-linear symmetric forms can be represented in a form $$ A = \sum_{k\ge 1} \lambda_k \varphi_k^{\otimes n}\ ?$$

(There is no compactness notion for multilinear forms, but we can assume that they are e.g. Hilbert--Schmidt.)