How to check diagonalization of orthogonal matrix? Is an orthogonal matrix diagonalizable over  $\mathbb{R}$ or $\mathbb{C}$ ?

I know that if all eigenvalues of a matrix are distinct then it is diagonalizable.
G.m. of each eigenvalue = its A.m.
And so on but sometimes there are complex matrices and I'm unable to find eigenvalues.
Please help me.
The matrix $\left(\begin{smallmatrix}0&amp;1\\-1&amp;0\end{smallmatrix}\right)$ is orthogonal, but it is not diagonalizable over $\mathbb R$ (it has no real eigenvalues). However, it is diagonalizable over $\mathbb C$. In fact, every real orthogonal matrix is diagonalizable over $\mathbb C$; this follows from the spectral theorem.
Actually, if we consider rotation matrix of order $2\times 2$, then we know rotation matrix is orthogonal matrix.then rotation matrix have no real eigen value in $\mathbb{R}$, unless the rotation angle $A$ is either $0$ or  $\pi$.
That is  rotation matrix have no eigenvector. 
Again $A$ linear operator on a real vector space need not have an eigenvector. 
That is, every  orthogonal matrix is not always diagonalizable over $\mathbb{R}$, but it is always diagonalizable over $\mathbb{C}$,because every linear operator on a complex vector space has at least one eigenvector.