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Rotations in 4 dimensions are performed around a fixed plane, they can be described by $SO(4)$, which is a group of orthogonal matrices with determinant equal to 1. It is easy to derive rotation matrices around the coordinate planes in $\mathbb{R}^4$, for example, $$ \begin{pmatrix} \cos(\theta) & \sin(\theta) & 0 & 0 \\ -\sin(\theta) & \cos(\theta) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$ performs a rotation around $X_3X_4$-plane by an angle $\theta$. What I am interested is are rotations around an arbitrary plane in $\mathbb{R}^4$. Given such a plane, how to get a rotation matrix for this plane?

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I notice you were careful to put "a rotation matrix." I really have to ask: was it intentional because you knew there is really more than one such matrix? :) –  rschwieb Mar 18 '13 at 20:43
    
Thanks for you answer. I knew that there is more than rotation matrix. –  Jimmy R Mar 18 '13 at 21:14

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Let $v,w$ be an arbitrary orthonormal pair of vectors in that plane. Extend it to an orthonormal basis of $\Bbb R^4$. Then the matrix that you gave above is a rotation matrix in terms of that basis.

To express it in another basis, just perform a change of basis to your desired basis.

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