I begin my question with a multilinear question then I will consider two local smooth analogies:

Assume that $\alpha$ is a real valued symmetric $k$-tensor, that is a $k$-linear map $\alpha:\overbrace{\mathbb{R}^n\times\mathbb{R}^n\times\ldots\times \mathbb{R}^n}^{k\; \text{times}} \to \mathbb{R}$ which is unchanged under permutations.

Is there an (orthonormal) linear map $T$ on $\mathbb{R}^n$ such that $T^*(\alpha)$ is in the diagonal form $T^*(\alpha)=\sum_{i=1}^n \lambda_i \overbrace{dx_i\otimes dx_i\otimes \ldots \otimes dx_i}^{k \;\text{times}}$

We would like to consider two kind of smooth analogies of this multilinear question as follows:

Assume that $\omega$ is a smooth symmetric $k$ tensor on an open set of $\mathbb{R}^n$:

What kind of obstructions would appear to represent $\omega$ in the form $\omega=\sum \overbrace{df_i \otimes df_i \otimes \ldots \otimes df_i}^{k \;\text{times}}$ for some smooth functions $f_1,f_2,\ldots,f_n$. When $k=2$ then this tensor is the pull back of the standard flat metric. Hence non vanishing of "Curvature" would be an obstruction for such a representation. What kind of tensors play the same role as curvature for $k>2$?

What kind of obstructions would appear to have a change of coordinate $\phi$ such that $\phi^*{\omega}=\sum_{i=1}^n g_i \overbrace{dx_i\otimes dx_i\otimes \ldots \otimes dx_i}^{k\;\text{times}}$ for some smooth functions $g_1,g_2,\ldots, g_n$.