Lie algebra of Spin Group

Question

In Peskin's QFT textbook, eq.(3.23): \begin{equation} S^{\mu\nu}=\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}] \end{equation} gives a Lie algebra representations of Lorentz group or more generally $SO(p,q)$, but author didn't mention how to deduce it. It is also known that the double cover of $SO(p,q)$ is spin group $Spin(p,q)$ whose Lie algebra is the same as that of $SO(p,q)$. My question is, does eq.(3.23) in the book actually form the Lie algebra representation of $Spin(p,q)$? If so, can one provide some details about it, or some references which mention the above result?