Is any proper open cone in $\mathbb{R}^n$ of the form $$C(x, e, \alpha) = \left\{ y\in\mathbb{R}^n : \arccos \frac{e\cdot(y-x)}{|y-x|} < \alpha\right\}$$ for unit vector $e$, angle $0<\alpha<\pi$ and vertex $x$, and uniquely represented by those $3$ values?
I'm asking this for further reference as well as to understand cones.
Edit: I realize now there are other types of open convex cones, so to make the question more meaningful I'm changing my question a little bit.
If $C_0$ is a non-empty open convex cone, does there always exist $C(x, e, \alpha)\subseteq C_0$?