In $\mathbb{R}^2$, all orthogonal matrices are one of two forms:
$$\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \text{ or } \begin{pmatrix} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{pmatrix} $$
for some angle $\theta \in \mathbb{R}$. In $\mathbb{R}^3$, all orthogonal matrices are of the form:
$$\begin{pmatrix} \cos\alpha\cos\gamma - \sin\alpha\sin\beta\sin\gamma & -\sin\alpha\cos\beta & -\cos\alpha\sin\gamma-\sin\alpha\sin\beta\cos\gamma \\ \cos\alpha\sin\beta\sin\gamma+\sin\alpha\cos\gamma & \cos\alpha\cos\beta & \cos\alpha\sin\beta\cos\gamma-\sin\alpha\sin\gamma \\ \cos\beta\sin\gamma & -\sin\beta & \cos\beta\cos\gamma \end{pmatrix}$$
where $\alpha,\beta,\gamma \in \mathbb{R}$ are angles.
How can I parametrize any orthogonal matrix in $\mathbb{R}^n$ using some number of angles, and the $\sin$ and $\cos$ functions?
Is there a recursive formula?