I have a rotation matrix
$$ R_{(\phi)} = \left( \begin{matrix} \cos (\phi) & \sin (\phi) & 0 & 0 \\ -\sin(\phi) & \cos(\phi) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} \right). $$
Now I needed to show that this rotation is in $SO(4)$. I showed that $\det R_{(\phi)} = 1$. Is that enough or are there more things that need to be shown here? Further I should compute the generator in this matrix (at $\phi=0$):
$$L = \frac{d}{d \phi} R_{(\phi)} = \left( \begin{matrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} \right) .$$ Now I need to find the basis of $SO(4)$, it's dimension and describe it in form of matrices. I have no idea how to do that. Does someone have a hint?