Addition in Geometric and Exterior Algebra

Question

I was trying to define in a mathematically precise way the constructions of Geometric Algebra does in terms of exterior algebra, and I found myself stuck in the following problem:

If one tries to construct the 2-dimensional algebra, $\Lambda^2$ from the vector space of one dimension less, $\Lambda^1$, or $V$, one should be able to write something like

$\Lambda^2 = \{ (x,y) | x,y ~ \in ~ \Lambda^1 \} $

However, I realized that not all elements of $\Lambda^2$ can be written as ordered pairs of elements of $\Lambda^1$. For example

\begin{equation} e_1 \wedge e_2 + e_3 \wedge e_4 \quad (1) \end{equation}

is an element of $\Lambda^2$ but is not factorizable as the product of two elements of $\Lambda^1$.

So my questions are:

1. What is a precise definition/construction of $\Lambda^2$ from $\Lambda^1$?
2. Could you give me a general, precise, definition of how to construct the complete Geometric Algebra of a space starting from $\Lambda^1$ (or the dimension/signature of the space)?
3. Since the addition in (1) cannot be made component-wise as usual, how do you define that addition operation?

Thanks for your answers and time.