How is quotient of the tensor algebra calculated to form the exterior algebra

Question

As stated on wikipedia and elsewhere the exterior algebra is the quotient of the tensor algebra by the ideal generated by $x \otimes y + y \otimes x$. But nowhere is this calculation given explicitly and I don't manage to do it myself. In dimension two there should be the set of the ideal that consists of all prefixes and postfixes to the generators. This is a countably infinite set. Then there should be three parallel cosets that are countably infinite as well. These four sets than form the exterior algebra. One set is the scalars, two form the vectors and one is the bivector. Please explain these sets and their generalisation to higher dimensions.