I want to refer to a mathematics book that explains the n-dimensional rotation matrix or rotation transformation.
Wikipedia concentrates most on 2D or 3D. There are things that one can say definition here and there, but I think it is not a good idea to use the definition there. Actually they don't seem to be definitions.
Strang's "Linear Algebra", Barret O'neill's "Elementary Differential Geometry" deal only with 2D or 3D cases. I think physicist are more interested in the general case, due to the theory of relativity. I found one explanation in "Geometrical Methods of Mathematical Physics" by Bernard Schutz. But I think it doesn't define the rotation matrix.
Artin's "Geometric Alegebra" defines the rotation group as an isometry $\sigma:V\to V$ such that $\det\sigma=1$. But the language there is so abstract that I can't catch any of them.
Can anyone give a reference that defines rotation transformation on $\mathbb R^n$ and state as a property that $A$ is a rotation matrix if and only if $A\in SO(n)$?
This is the end of the question and the below is what I wanted to do. I wanted to prove that if $A\in SO(n)$, then $A$ is a rotation about a line through the origin in $\mathbb R^n$. So I need to define the rotation transformation(or matrix) in $n$ dimensional Euclidean space.