Non commutative multiplication of two vectors

I'm trying to implement the axioms for geometric products of two vectors in n-dimensioned Euclidean vector space.

i.e Allowing the user to input the number of dimensions, for e.g 3

Generate e1,e2,e3 as symbols

Allow the user to input two vectors as functions of n-sized vectors eg a=a1e1+a2e2+a3e3; b=b1e1+b2e2+b3e3

where a1,a2... and b1,b2... are scalars

Expand the two vectors a and b

(a1e1+a2e2+a3e3)(b1e1+b2e2+b3e3)

It all works fine for me until this point. From this point, I require the expansion operation to be non commutative, i.e e2e1=/=e1e2

Is there a way I can do this.

Following that, if the reader is feeling kind, is there a way to implement geometric algebra axioms such as e1e1=e2e2=enen=1

and e2e1=-e1e2 -> eiej=-ejei

Thanks a lot!