The Lorentz group is the group of $4\times 4$ matrices that satisfy $$\Lambda^T \eta \Lambda = \eta, \eta = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$$ Is there a way to parametrize this group, just like how we may parametrize the group of 3-dimensional rotations (=$SO(3)$) using an axis of rotation and the angle of rotation?
Wikipedia has a formula for a Lorentz transformation with a boost along a direction $v$. Perhaps it's possible to write all Lorentz transformations in this way?