What are the new coordinates of P after rotation?

Question

Let $P=(x_1,x_2,...,x_n)$ and you want to rotate around the $x_1$ axis by $\theta$, what are the new $P'= (x_1',x_2',...,x_n')$ coordinates?
In 2D and 3D it's easy, because we can write up the matrix for rotating and just take the matrix product.

Answer 1

$\begin{bmatrix} 1 & 0 & 0 &... & 0 & 0 & 0 \\ 0 & 1 & 0 &... & 0 & 0 & 0 \\ 0 & 0 & 1 &... & 0 & 0 & 0 \\ &&& ... \\ 0 & 0 & 0 &... & 1 & 0 & 0 \\ 0 & 0 & 0 &... & 0 & \cos{\theta} & \sin{\theta} \\ 0 & 0 & 0 &... & 0 & -\sin{\theta} & \cos{\theta} \\ \end{bmatrix}$

is the associated matrix for the linear transformation that rotates the $x_{n-1}, x_n$ plane an angle $\theta$ counterclockwise. Rotation of other planes would have the same Jordan canonical form, although the matrix may appear different