Vector product for higher dimensions than 3?

I know that the vector product (for $\displaystyle \Bbb{R}^3$) is defined in a way so that a orthogonal vector is produced from two original vectors. In $\displaystyle \Bbb{R}^2$ you can't create a vector $\displaystyle \neq\overline{0}$ orthogonal to two linear independent vectors. On the other hand, you can if you only have one vector from the beginning. In $\displaystyle \Bbb{R}^4$, you can if you have 3 linear independent vectors from the beginning (you'll get a line of possible vectors contrary to if you only have 2 vectors to perform the multiplication with, then you'll get a plane).

Is there some kind of general vector product for $\displaystyle n-1$ vectors in $\displaystyle \Bbb{R}^n$? (this would be almost the same as a method for obtaining a vector orthogonal to the other vectors.)