Combining, we have , and so as desired. If is continuous and each is a basis for , show that . The function is a continuous function, whose image does not contain 0 since is a basis for every t. By the intermediate value theorem, it follows that the image of consists of numbers all of the same sign. So all the have the same orientation.
Substitution shows that works.
If are linearly independent, show that is the usual orientation of . By the definition, we have . Since the are linearly independent, the definition of cross product with completing the basis shows that the cross product is not zero. So in fact, the determinant is positive, which shows the result. Show that every non-zero is the volume element determined by some inner product and orientation for .