Let $V$ be an $n$-dimensional vector space.
The exterior algebra $\Lambda V$ of $V$ is the direct sum of the exterior powers $\Lambda^kV$. It comes with a product (called the exterior product) which is bilinear, alternating and anticommutative. The dimension of $\Lambda V$ is $2^n$.
(For physics buffs, the exterior algebra is an example of a supersymmetric and supercommutative algebra.)
The exterior algebra is a geometric algebra with trivial quadratic form (I think). It is a quotient of the tensor algebra of $V$ (by the two-sided ideal generated by set $\{v \otimes v\ |\ v\in V\}$).
The exterior algebra $\Lambda\textbf{R}^1$ of $\textbf{R}^1$ is isomorphic to the dual numbers.
What is the exterior algebra $\Lambda\textbf{R}^2$ of $\textbf{R}^2$ (as in, through which isomorphic objects can I understand it, specially for the purpose of doing explicit computations)? Is it related to $\Lambda\textbf{C}$? Does it admit a matrix representation, like $\Lambda\textbf{R}^1$?
(If this question is too low-powered please do migrate it / close it.)