Define the orthonormal frame of a space $R^n$ to be a set of vectors ($b_1$, $b_2$, ..., $b_n$) if these vectors together form an orthonormal basis of $R^n$, and denote this frame to be $F_n$.
Similarly, define positively oriented orthonormal frames to be $F_{n}^{+}$, just that this time only positively oriented bases are considered.
How do you identify $F_n$ with orthogonal group $O(n)$, and how do you identify $F_{n}^{+}$ with special orthogonal group $SO(n)$?
I a bit confused how far should this "identification" goes. For example, for the first part of question, if I can just pick a random orthogonal matrix from the given $O(n)$ group, then I can get a set of orthogonal basis of $R^n$, and finally normalize it to get the orthonormal version? Since there is no extra requirements on which specific orthonormal frame I should get?
For the latter part of question I am indeed confused on how to get $F_{n}^{+}$ from $SO(n)$.
Thanks for suggestions.