For a $k$-linear operator $T \colon \mathbb{R}^n \times \cdots \times \mathbb{R}^n \to \mathbb{R}$ its norm can be defined as: \begin{equation} \lVert T \rVert = \sup_{ \lVert x_i \rVert_{\mathbb{R}^n} \le 1 \text{ for } i \in \{1, \ldots, k\} }\lvert T(x_1, \ldots, x_k) \rvert. \end{equation}
If $k=2$ and $T$ is symmetric, i.e. the order in which we put the arguments in $T$ does not change its value, then we can use alternative definition: \begin{equation} \lVert T \rVert = \sup_{ \lVert x \rVert_{\mathbb{R}^n} \le 1 } \lvert T(x,x) \rvert. \end{equation}
Question
Suppose now that $T$ is symmetric $k$-linear for some $k>2$. Do we have a similar alternative norm (which might only be equivalent but not necessarily equal to the usual k-linear functional one)?