Orientation on finite dimensional vector spaces over finite fields.

Solution 1:

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.