For those who are interested, this sort of algebra would be known as the [Grassman algebra or the exterior algebra](https://en.wikipedia.org/wiki/Exterior_algebra). It becomes much more interesting if you use non-orthonormal bases (or non-euclidean geometry), since then you need to introduce a dual basis and distinguish between contravariant vectors and covariant vectors. When you add derivatives to the mix you end up in differential geometry.
Yes, this is exterior algebra. It's also interesting to figure out how this works in ambient dimensions other than three. The author has a table of grades: 0 for scalars, 1 for vectors, 2 for "bivectors", 3 for "trivectors", and they count the number of bases for each of these grades as 1 3 3 1. These basis counts are the dimensions of the (vector space of) scalars, vectors, "bivectors", "trivectors". If you go to two ambient dimensions you get 1 2 1, and if you go to four ambient dimensions you get 1 4 6 4 1. It's Pascal's triangle.
Grassmann algebra is a very important part of it, in fact you can reconstruct it in geometric algebra. More generally though, this algebra would be known as Clifford algebra.
