Not all hypersurfaces in $\mathbb{R}^n$ are of contact type.

Weinstein, in the paper: "On the hypothesis of Rabinowitz' periodic orbit theorems", where he defined the concept of contact type, gives a criterion. If $H^1(\Sigma)=0$, and $\Sigma$ is of contact type, then the characteristic line bundle comes with a distinguished orientation, determined by those vectors $\xi$ such that $\alpha(\xi)>0$ for all contact forms $\alpha$. This is independent of the contact form. For periodic orbits this induces a positivity criterion. In the same paper he also constructs an hypersurface in $\mathbb{R}^4$ which is not of contact type, by showing that the criterion is violated.

Many hypersurfaces are of contact type, as you remarked. Another nice example are *mechanical hypersurfaces*. These are hypersurfaces arising as level sets from hamiltonians $H=T+V$, where $T$ is the kinetic energy term $T=1/2|p|^2$, and $V$ is a potential depending only on $q$. This works in general cotangent bundles.