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It is well known that the $2 \times 2$ rotation matrix is given by, $$\left[ \begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \\ \end{array} \right]$$ and that there are $3 \times 3$ rotation matrices to describe rotation in 3-dimensions,

$$R_{x}(\theta)=\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) \\ 0 & \sin(\theta) & \cos(\theta) \\ \end{array} \right]$$ $$R_{y}(\theta)=\left[ \begin{array}{ccc} \cos(\theta) & 0 & \sin(\theta) \\ 0 & 1 & 0 \\ -\sin(\theta) & 0 & \cos(\theta) \\ \end{array} \right]$$ $$R_{z}(\theta)=\left[ \begin{array}{ccc} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \\ \end{array} \right]$$ That is, there are three elements of the rotation group $SO(3)$, and there is one element of $SO(2)$. In general, I found that for $\phi$ elements of $SO(d)$, where $d$ is the dimension that $$\phi = \frac{d(d-1)}{2}$$ Yet, how is it that the explicit representation of say $R_{i}(\theta)$ is derived, where $i$ is some arbitrary element of $SO(d)$. Are these things all manually computed or is there a general formula/method for determining what they are? The reason I am asking is that I have a functional in configuration space $T$ that depends on a parameter $R_{j}^{i}$ that is summing over dimensions $i,j$. $R_{j}^{i} \in SO(d)$ is representing an $i \times j$ rotation matrix (note $d=i=j$) and I am having trouble explicitly constructing an $n$-dimensional example since I don't know how to represent an $n \times n$ rotation matrix. Any references to papers, or original responses are welcome.

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Exponentiate a skew-symmetric matrix: en.wikipedia.org/wiki/Skew-symmetric_matrix –  Qiaochu Yuan Jan 18 '12 at 1:12
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The generalization of $R_x(\theta)$, $R_y(\theta)$, and $R_z(\theta)$ to $n$ dimensions are the Givens rotations. –  Rahul Jan 18 '12 at 1:33
    
@RahulNarain: Could you give an explanation so I can accept it as an answer? The wikipedia page is blocked because of the SOPA act awareness thing. –  Samuel Reid Jan 18 '12 at 5:51

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