This is a great question.
Have you seen Manin’s book “Cubic forms: algebra, geometry, arithmetic”?
A form of degree greater than $2$ is a homogeneous polynomial of degree greater $2$. Of course there are important examples of such polynomials (e.g., the determinant polynomial of degree $n$), but a major distinction between $n=2$ and $n>2$ is that orthogonal groups of quadratic forms can be quite intricate (rotations, reflections, etc), while in higher degree the linear transformations preserving a form $f$ — its “orthogonal group” - form a finite group if $f$ has degree at least $3$, nonzero discriminant, and we are working over very a field of characteristic $0$ or characteristic bigger than $n$ (theorem of Jordan classically, meaning over the complex numbers).
The sum of squares quadratic form is a very rich object algebraically and geometrically, due to its big orthogonal group. But over $\mathbf C$, the only linear transformations preserving $\sum_{i=1}^n x_i^n$ for $n>2$ are compositions of coordinate permutations and scaling of coordinates by $n$th roots of unity: a finite group of order $2^n n!$. Quite the contrast.
For $n>2$, forms of degree $n$ often can’t be diagonalized.