Rotation matrix in $n$ dimension along $n$ dimensional vector

Question

I want to construct a Rotation matrix in general $n$ dimension along $n$ dimensional vector $vec{p}$.
First I know how to construct $n=3$ case : consider $n=3$ rotation along $3$ dimensional vectork $vec{k}$, then
Rotation matrix $R$ is given by
begin{align}
mathbf{R} = mathbf{I} + sin(theta) mathbf{K} + (1-cos(theta)) mathbf{K}
end{align}
where
begin{align} mathbf {K} =left[{begin{array}{ccc}0&-k_{z}&k_{y}k_{z}&0&-k_{x}-k_{y}&k_{x}&0end{array}}right] 
end{align}
How about a general $n$ dimensional case?
Is a similar formula exists? then What is it?

Richard Myers · Accepted Answer

I suspect a close-form formula may be hard to come by. However, there is a general prescription for constructing such matrices, though actually doing the construction by hand may not be possible.
Taking as definition that by "rotation" we mean a matrix which leaves the inner product of two vectors $x,y$ invariant:
$$
x^{prime T}y^prime=x^prime R^TRy^prime=x^Ty,
$$
where $x^prime = Rx$ for some rotation matrix $R$, and similarly for $y^prime$. Such matrices are known as the defining representation of the group $SO(n)$ where the dimension of the vectors is $n$.
There are general results about Lie groups (of which $SO(n)$ is an example) which tell us that there exist so-called generators of the group which when exponentiated give the elements of the group:
$$
R(theta)=e^{theta^aomega_a}
$$
where $theta^a$ are the parameters determining the rotation (for example, the Euler angles in $n=3$) and the $omega_a$ are some collection of yet-to-be-determined matrices.
Now, we took $R^TR=I$ to be the determining relation for our rotations. Let's expand this requirement to first order in $theta^a$ to find
$$
1=(1+theta^aomega_a^T)(1+theta^aomega_a)=1+theta^a(omega_a^T+omega_a).
$$
Since this must be the case for all (small values of) $theta^a$, it must be the case that each of the generators $omega_a$ must satisfy
$$
omega_a=-omega_a^T,
$$
and hence they must all be antisymmetric matrices.
So the result of all this is: in order to generate a rotation in $n$ dimensions, pick some linear combination of antisymmetric $ntimes n$ matrices and exponentiate it.
Edit: Let me also add, because I notice you also asked specifically about rotating about a given vector: in order to demand that a given vector $v$ be left invariant by the rotation (so the rotation is about $v$), we are demanding
$$
R(theta)v=v
$$
and hence, expanding again to first order in $theta$, we must have
$$
(1+theta^aomega_a)v=v.
$$
This necessitates $theta^aomega_a v=0$. So the collection of rotations which leaves a given vector invariant (rotates about that vector) will be all those rotations formed by an antisymmetric matrix (the linear combination $theta^aomega_a$) which contain $v$ in the kernel.

Rotation matrix in $n$ dimension along $n$ dimensional vector

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