Is the cross product of two vectors a vector? Yes and no. Yes, of course, $\uu\times\vv$ is a vector, but is it the same type of vector as $\uu$ and $\vv$? Not really.

First of all, the dimensions are different: If $\uu$ and $\vv$ have dimensions of length, then $\uu\times\vv$ has dimensions of area. This behavior arises from the geometric definition of the cross product as directed area.

A more interesting difficulty arises when considering symmetries. What happens if one reflects points through the origin, a so-called parity transformation? Each vector $\uu$ would transform into $-\uu$. But $\uu\times\vv$ would transform into itself, since $(-\uu)\times(-\vv)=\uu\times\vv$!

There thus appear to be two types of vectors, which were historically called vectors and pseudovectors, respectively. Similarly, there are two types of scalars, called pseudoscalars and scalars, depending on whether they do, respectively do not, change sign under a parity transformation.

Now recall that over (Euclidean) $\RR^3$ we have two 3-dimensional spaces of forms (the 1- and 2-forms), and two 1-dimensional spaces (the 0- and 3-forms). How do forms behave under a parity transformation? Each basis 1-form transforms into its negative, so odd-rank forms pick up a minus sign, whereas even-rank forms remain the same. With the wisdom of hindsight, it is now easy to identify not only 0-forms as scalars and 1-forms as vectors, but also 2-forms as pseudovectors and 3-forms as pseudoscalars.