Is an Exterior Product the "Opposite" of an Inner Product One is represented by a dot product, the other by a cross product. The "inner product collapses two co-ordinate vectors into a scalar, the exterior product seems to expand them in a multilinear (manifold)? The inner product seldom has "cancellation," the exterior product has a lot of cancellation (between the same differential forms).
Are they just two opposite sides of the same coin? If not, why do they seem to be so "parallel"?
 A: Don't make to much from the words "inner", "interior" and "exterior"; none of them is the opposite to another of them. It is a fact that in connection with a vector space (maybe provided with a scalar product) there are various natural bilinear (or multilinear) functions.
The first of them (called "opération extérieure" by Bourbaki) is the map $F\times V\to V\ $  ($F$ being the ground field) $(\alpha, x)\mapsto \alpha x$. If the ground field $f$ is $\ ={\mathbb R}$ or $\ ={\mathbb C}$ then $V$ can be provided  with a ${\it scalar\ product}\ $ $\bullet:\ V\times V\to F$. This scalar product is a symmetric bilinear function with the extra property that $x\bullet x >0$ for all $x\ne 0$ and is often called the ${\it inner\ product}\ $ on $V$. 
Independently of $\bullet$ one can set up the following: If $F$ is any ground field of characteristic $\ne2$ then there is a certain algebraic construction called ${\it exterior\ product}\ $ $\wedge$, which is skew-multilinear on the cartesian powers $V^r$, $\ 1\leq r\leq{\rm dim}(V)$. 
Now combine the two constructions: If $V$ is a real vector space provided with a scalar product, and if ${\rm dim}(V)=3$, then any parallelepiped spanned by three vectors $p$, $q$, $x\in V$ has a natural volume $\mu(p,q,x)$. The volume form $\mu(\cdot,\cdot,\cdot)$ is trilinear and gives negative values for "negatively oriented" triples $p$, $q$, $x$; it is uniquely determined by the condition that the volume of an a priori chosen orthonormal basis $(e_1,e_2,e_3)$ should be $=1$. Given this volume form $\mu(\cdot,\cdot,\cdot)$ the exterior  product $p\wedge q$ of two vectors $p$, $q\in V$ can be identfied with a unique vector $r\in V$ where $r$ is attached to $p$ and $q$ by the identity
$$\mu(p,q,x)=r\bullet x\qquad(x\in V)\ .$$
This vector $r$ is then called the vector product or the exterior product of the given vectors $p$ and $q$.
