Probably a very simple question: Given the standard Cartesian coordinate matrix, $$\begin{pmatrix}1 & \\ & 1 & \\ & & 1\\ & & & 1\\ & & & & 1\\ & & & & & .\\ & & & & & & .\\ & & & & & & & .\\ & & & & & & & & 1 \end{pmatrix}$$
what is the matrix which is the extension of "basic rotations" to n-dimensions, as shown [in Wikipedia] for 3 dimensions.
And say we wish to rotate by some arbitrary order around several coordinates $k$ by $\theta$ - would the rotation matrix then simply be the multiplication of such $k$ matrices as above?