Let's presume I have an arbitrary plane, for sake of simplification, centered at (0,0,0), described by coordinates of 4 vertices (and normal if needed).
Is there any way to describe this plane as rotation of the same plane centered at (0,0,0), lying on XY plane. With any possible way of describing 3D rotation, such as Euler angles, axis angle or quaternion, any of them works.
If they're both centred at (0,0,0), it is sufficient to use the (unit) normal vector to parameterize them. The reference plane has normal vector $(0,0,1)$, the other guy's normal vector is generally of the form $(\cos\theta \sin \phi, \sin\theta\sin\phi,\cos\phi)$.
This tells you you need a rotation about $y$-axis by $\theta$, followed by a rotation about the $z$-axis by $\phi$.
(Note that you can calculate these angles given some vector $(x,y,z)$ by solving the equation $(x,y,z)=(\cos\theta \sin \phi, \sin\theta\sin\phi,\cos\phi)$ over the domains $0\leq \theta \leq 2\pi$ and $0\leq \phi \leq \pi$).