SO(n) and O(n) questions

[James]

Dear all,

I am trying to understand more the groups SO(n) and O(n) geometrically.  I 
think I have a good understanding of them algebraically.

Well, my first question is : What does the word "rotation" mean in R^n?  In 
R^3, the map A : R^3 ---> R^3 is a rotation if it preserves distances between 
points, fixes the origin, fixes some non-zero vector, and represents a 
2-dimensional rotation in the plane perpendicular to this fixed vector.

So what does "rotation" mean in R^n?  Then, why does SO(n) give all rotations 
in R^n?  I know why SO(3) gives all rotations in R^3 (proof in Artin's algebra 
page 129).  Also, geometrically, what is O(n)?  I would like to learn even more 
about SO(n) and O(n) geometrically, but I am not sure which questions to ask.  
Can you recommend a good reference for me to learn?


Sincerely,

James