What are the purposes of $k$-blades? Why is it important to have a oriented area, or an oriented volume? I'm referring to $k$-blades in such a way that $$\hat{v_1} \wedge \cdots \wedge \hat{v_k} $$ produces a $k$ dimension oriented figure.
What purpose do these serve?
For the same reason that it's important to have vectors, which are "oriented lengths." They're part of the natural setting for differential geometry.