In some ways, the vector product is a feature unique to three-dimensional space. But there is a construction called the wedge product that generalises some of the properties of the vector product to higher-dimensional spaces.
I know that the vector product (for ) is defined in a way so that a orthogonal vector is produced from two original vectors. In you can't create a vector orthogonal to two linear independent vectors. On the other hand, you can if you only have one vector from the beginning. In , 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 vectors in ? (this would be almost the same as a method for obtaining a vector orthogonal to the other vectors.)