Let $\mathbb{R}^n$ be the Euclidean space of dimension $n.$
We say that two ordered bases have the same orientation whenever the change-of-base matrix has determinant $>0.$
This determines an equivalence relation. Moreover, we say that the standard basis is positively oriented, so that any base equivalent to the standard basis under this relation is positively oriented.
I would like to understand what does this mean geometrically. A change-of-base matrix is a symmetry $g\in \text{Aut}(\mathbb{R}^n)=\text{GL}(n,\mathbb{R})$ and we define two ordered bases to be oriented in the same way whenever the unique symmetry realizing the change of base belongs of the subgroup of $\text{GL}^+(n,\mathbb{R})$ of $\text{Det}>0$ matrices. What is the geometric interpretation of this group?
Is there such a thing as an orientation of a single vector, so that two bases are oriented in the same way if and only if the change of base symmetry does not change the orientation?
Or maybe the choice of a basis prescribes the same orientation to all vectors? That is, expressing a vector with respect to a positive base means choosing a positive orientation of the vector? So that once a basis is chosen, all vectors are oriented in the same way?