Given a finite-dimensional real vector space V, is there a total order O on the vectors in V such that any basis of V, when ordered according to O, has the same orientation?
Does existence depend on the dimensionality of the vector space? If so, in which way?
Whenever such an order exists, how can (a simple) one be constructed?
If there is no such order, I would be interested in any generalization: for example a simple rule to order any basis in V such that it always leads to the same orientation.