To give something explicit in sine and cosine,
$$
\left( \begin{array}{ccc}
\cos\theta \cos\psi & -\cos\phi \sin\psi + \sin\phi \sin\theta \cos\psi & \sin\phi \sin\psi + \cos\phi \sin\theta \cos\psi \\
\cos\theta \sin\psi & \cos\phi \cos\psi + \sin\phi \sin\theta \sin\psi & -\sin\phi \cos\psi + \cos\phi \sin\theta \sin\psi \\
-\sin\theta & \sin\phi \cos\theta & \cos\phi \cos\theta
\end{array}
\right)
$$
Note that three parameters are required. In odd dimension, there is a real eigenvalue. For $SO_n$ this eigenvalue is $+1.$ So there is a fixed vector in some direction. It takes two parameters to specify this point on the unit sphere. The Lie group element is then a rotation around this point. So it takes a third parameter specifying the amount of rotation about that axis.
