Given a quadratic form $Q(x_1, ..., x_n)$, there is a natural notion of rank defined by looking at the rank of the unique symmetric matrix associated to the quadratic form, i.e. we consider the symmetric matrix $A$ such that $\mathbf{x}^T A \mathbf{x} = Q(\mathbf{x})$.

Suppose we have a $F(\mathbf{x})$, a degree $d$ homogeneous form. I naively thought that perhaps the rank of $F$ should be defined to be the rank of a hypermatrix $A$ attached to $F(\mathbf{x})$ (Assuming such notions makes sense). But I have never seen anything like that in the literature. So perhaps there are issues if one wants to define a rank of forms this way.

I was wondering if someone could explain me why considering this may not make sense or not considered? Thank you very much!