This is correct. Another common terminology is that a vector is a 1-form, and a bi-vector a 2-form. The wedge of an n-form and an m-form is an (n+m) form, whic

This is correct. Another common terminology is that a vector is a 1-form, and a bi-vector a 2-form. The wedge of an n-form and an m-form is an (n+m) form, which generalises the cross product. To define the dot product you need the notion of a Hodge star, which maps an n-form to a (D-n)-form where D is the dimension of the ambient space. If you learned linear algebra, or especially vector calculus, without learning these things then you were cheated out of the best bits, and I encourage you to rectify this! The general picture is actually clearer than the 2D or 3D one usually taught, in which some things happen to co-incide. https://en.wikipedia.org/wiki/Differential_form https://en.wikipedia.org/wiki/Differential_form isn't amazing but will give you an idea what to google.