An ordered vector basis for a finite-dimensional vector
space
defines an orientation. Another basis gives the same orientation if the matrix has a positive determinant, in which case the basis is called oriented.

Any vector space has two possible orientations since the determinant of an nonsingular
matrix is either positive or negative. For example, in , is one orientation and is the other orientation. In three
dimensions, the cross product uses the right-hand
rule by convention, reflecting the use of the canonical orientation as .

An orientation can be given by a nonzero element in the top exterior power of ,
i.e., .
For example,
gives the canonical orientation on and gives the other orientation.

Some special vector space structures imply an orientation. For example, if is a symplectic form
on ,
of dimension ,
then
gives an orientation. Also, if is a complex vector space,
then as a real vector space of dimension , the complex structure
gives an orientation.