I know the construction of the Hodge star operator in the context of (pseudo-)euclidean real vector spaces. Apart from the scalar product it involves a orientation of the vector space, which one has to choose (at least if one is not willing to deal with forms of odd parity). 


While I was wondering how to extend this definition to vector spaces over arbitrary fields, I realised that I even do not know what an orientation of a vector space $V$ over a general field $k$ is. In case there is such a notion: How could one use the definition in order to define a Hodge star operator?) Or is there a better definition?


A slight extension of this question I am also interested in is the case of a finitely presented flat module over a local ring (which includes the case where the module is the germ of sections of a vector bundle over a locally ringed space at a point).



How to define the orientation of a vector space over an arbitrary field?
An orientation of the $n$-dimensional real vector space $V$ is an equivalence class of generators of the $1$-dimensional vector space $det(V)=\Lambda^n(V)$ under the relation $\omega\sim c\omega$, $c>0$.


A basis-free description of the usual Hodge star for a real vector space with positive inner product is exactly as Mariano has described it (up to some sign depending on $k$ and $n$, I believe). This description can be used to define a "Hodge star" for any finite-dimensional $V$ over any field, as soon as an inner product and a volume are given. 


But in the usual (real, positive) case we also require the volume to be compatible with the inner product in the sense that $\langle\omega,\omega\rangle=1$, following a standard convention for extending an inner product on $V$ to exterior powers of $V$. A positive definite inner product on $V$ gives a positive definite inner product on $det(V)$, and there are exactly two possible compatible choices of $\omega$, one for each orientation of $V$.


Without this compatibility, the star that you get will not have the usual formal properties. The scalar $\langle\omega,\omega\rangle$ will enter into the formula for star composed with star.


For other fields, or even for indefinite inner products in the real case, there will not always be a compatible $\omega$, but if there is one then there will be exactly two: the question of existence of a compatible volume for a given inner product has to do with $k^{\star}/k^{\star 2}$, while the uniqueness has to do with the kernel of squaring $k^{\star}\to k^{\star}$. (In the real case these groups are easily confused, since they are isomorphic, and in fact isomorphic via the obvious map.)


I suppose that in the real but not definite case one settles for using a volume for which $|\langle\omega,\omega\rangle|=1$; such a volume exists and is unique up to sign.


