A parametric question of orthogonal matrices [closed]

Suppose that $$U_{\theta }=\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$$,in which $$\theta$$ is a real parameter,it is easy to prove that a given matrix $$U\in M_{2}(\mathbb{R})$$ is real orthogonal matrix if and only if either $$U=U_{\theta}$$ or $$U=\begin{bmatrix} 0 &1 \\ 1 & 0 \end{bmatrix}U_{\theta}$$. But for 3-by-3 matrices, how to construct a parametric presentation of 3-by-3 real orthogonal group?

• Have you tried Euler angles? – Eric 2 days ago
• @Eric I don’t know how to use. Would you mind giving me some tips? thank you very much. – Stephen Wong 2 days ago
• I doubt you can get a neat formula but you could process in the same way as in the 2-dimensional case. That is, since U is orthogonal its rows (and columns) have length 1 and they are orthogonal to each other. Then solve the equations depending on two parameters. – Salcio 2 days ago
• @Salcio Thanks for your tips! – Stephen Wong 2 days ago
• Euler angles are effectively rotating around x axis, then z -axis, then x-axis again. You can compose those matrices. en.m.wikipedia.org/wiki/Euler_angles – Eric yesterday