Hi all. Is there any explicit matrix expression for a general element of the special orthogonal group $SO(3)$? I have been searching texts and net both, but could not find it. Kindly provide any references.



Matrix expression for elements of $SO(3)$
Here is the standard [quaternion](http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation) answer: Given $(a,b,c,d)$ such that $a^2+b^2+c^2+d^2=1$, the matrix
$$\begin{pmatrix}
a^2+b^2-c^2-d^2&2bc-2ad &2bd+2ac \\
2bc+2ad &a^2-b^2+c^2-d^2&2cd-2ab \\
2bd-2ac &2cd+2ab &a^2-b^2-c^2+d^2\\
\end{pmatrix}$$
is a rotation and every rotation matrix is of this form. Note that $(a,b,c,d)$ and $(-a, -b,-c,-d)$ give the same rotation.


