One sometimes useful decomposition of four-vectors uses two Euclidean angles and what is called the rapidity, a sort of unitless measure of energy.
You can imagine starting with a four-vector in its rest frame, $p=(m,0,0,0)$, where $p^2 = m^2$. Then you can boost in the $z$ direction with rapidity parameter $\eta \in [0,\infty)$, giving you $p = m(\cosh\eta,0,0,\sinh\eta)$. You can check that this left the Minkowski norm invariant. Finally, you can do a spatial rotation to make the three-vector part point any direction you want. $p = m(\cosh\eta,\sinh\eta \sin\theta\cos\phi,\sinh\eta\sin\theta\sin\phi,\sinh\eta\cos\theta)$.
For physical interpretation, note that $E = m\cosh\eta$, so $\cosh\eta$ is the same as the Lorentz factor $\gamma$, and $\sinh\eta = \beta\gamma$.
Note this form is only for time-like vectors. For lightlike vectors or spacelike vectors you have to start with different initial four-vectors, like $m(1,0,0,1)$ or $m(0,0,0,1)$.