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I want to be able to computationally generate a rotation matrix for $\mathbb{R}^n$ where $n$ might go as high as $10^4$. The naive technique would be to generate the rotation in each plane then proceed by matrix multiplication. This seems unwise.

Ignoring the special cases $n=1$, $n=2$; Can I generate the matrix of rotation about $\binom{n}{2}$ planes for $\mathbb{R}^n$ without using the naive technique?

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    Or $\binom n2$ angles for the axis-aligned planes. See http://en.wikipedia.org/wiki/Rotation_matrix#Decompositions.2012-09-30

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