For finite-dimensional $\mathbb R$-vector spaces, we define an orientation to be an equivalence class of ordered bases, where $B_1 \sim B_2$ iff the change of basis matrix $A$ taking $B_{1}$ to $B_{2}$ has positive determinant. Then there are two equivalence classes, one which we call, "positive" and the other we call, "negative".
I wanted to know if any work has been done to extend this to finite-dimensional vector spaces over finite fields. My idea was to do everything as above, but replace the condition $\det A>0$ with $\det A$ is a quadratic residue. This should split the bases into two equivalence classes just as above.
I played around with these and noticed some interesting things. For example let $q=p^{f}$ and $\mathbb F_{q}$ be an $\mathbb F_{p}$ vector space. Unlike the case for $\mathbb R$, if $q \not\equiv 3 \pmod 4$ then switching any two vectors in your basis doesn't change the equivalence class, because the determinant of the corresponding change of basis matrix is $-1$, which is a quadratic residue in this case.
Has any work been done on this, or are there any other definitions or related concepts that might be of interest?
The concept which generalizes to all fields is not an orientation but a "volume form," by which I mean a nonzero element of the top exterior power $\Lambda^n(V)$ of an $n$-dimensional vector space over a field $k$. When $k = \mathbb{R}$, the space of volume forms has two connected components (indeed it can be noncanonically identified with $k^{\times} = \mathbb{R}^{\times}$), and a choice of such a connected component gives an orientation. More generally, if $G$ is any topological group, looking at connected components gives a natural homomorphism $G \to \pi_0(G)$, so one can think about a choice of orientation as a choice of element in the image of the natural map $$\Lambda^n(V)^{\times} \to \pi_0 \left( \Lambda^n(V)^{\times} \right)$$
Over an arbitrary field you can just pick any quotient group of $\Lambda^n(V)^{\times}$ and consider the corresponding choice of "generalized orientation," e.g. above I suggested quotienting by the subgroup of squares. A choice of volume form will then always naturally give rise to a "generalized orientation."
But it's unclear whether this is of any use.