I recently came across the notion of multivectors and exterior algebras recently and while I find these notions to be very enlightening, I also feel a strong temptation to interpret these objects too geometrically (at least for my taste).
My current understanding is this:
Given a (finite dimensional) vector space $V$ over a field $\mathbb{F}$ and a choice of basis $\mathcal{B}=\{e_{1},...,e_{n}\}$ we can define the notion of orientation. If we let $V =\mathbb{R}^{2}$ then in some sense, we can think of a vector as a "measure" with a direction and this direction depends on a choice of basis (usually the canonical basis for $\mathbb{R}^{2}$). In a vector space $V$, we can add vectors $u,v \in V$ together to create new vectors $u+v \in V$ and multiply vectors by elements of $\mathbb{F}$ to create "new" vectors $\alpha u+ \beta v \in V$ for $\alpha,\beta \in \mathbb{F}$.
However, we cannot multiply vectors together to get another vector unless we define an algebra $\Lambda(V)$ over $\mathbb{F}$ which in addition to the properties of a vector space, gives us a new binary operation $\wedge:V \times V \to \Lambda(V)$ such that $$u \wedge v = -v \wedge u \in \Lambda(V)$$ and $$u \wedge (v \wedge w)=(u \wedge v) \wedge w=u \wedge v \wedge w \in \Lambda(V)$$ and furthermore we have $$ (\alpha_{1} e_{1} + \beta_{1} e_{2})\wedge(\alpha_{2} e_{1}+\beta_{2} e_{2})=\alpha_{1}\alpha_{2} e_{1} \wedge e_{1}+\alpha_{1}\beta_{2}e_{1} \wedge e_{2}+\beta_{1}\alpha_{2} e_{2} \wedge e_{1}+\beta_{1}\beta_{2}e_{2} \wedge e_{2} $$ which then gives us $$ (\alpha_{1}\beta_{2}-\beta_{1}\alpha_{2})e_{1} \wedge e_{2} $$ so $\Lambda(V)$ is anticommutative with respect to $\wedge$, $\wedge$ is associative and $\wedge$ distributes over (vector) addition.
Now my problem is with the geometric intuition\interpretation behind these objects. Basically, with a vector you are constrained to the geometric interpretation of "directed line segment". However, for $u,v \in V$, we have $u \wedge v \in \Lambda(V)$, which is no longer a vector but a bivector which seems like it can be interpreted as a "measure" with an orientation. If we take $V=\mathbb{R}^{2}$, we can interpret $u \wedge v$ as the area of the parallelogram in the picture 
However, it isn't the shape that is so important but the area and the orientation induced by the exterior product. So my understanding is that a bivector can be informally thought of as an equivalence class of regions in the plane with the same area and the same orientation. However, as a mathematician (in training) this amount of geometric reasoning doesn't sit right with me.
On one hand, I know that I just need to put the geometric aspects aside and think of $u \wedge v$ as an abstract object in $\Lambda(V)$ but I also fear that if I don't get the geometric intuition, I am going to miss out on the insights behind Clifford/Geometric Algebra down the road (should I ever venture there). I was hoping some of the more experienced mathematicians on here could give me some insight on the matter or any books (with a mathematical perspective rather than physical) that might help me see this area of mathematics clearer?