In https://docs.gap-system.org/doc/ref/chap61.html section 61.13-2 ExteriorPower one can find how to construct the exterior power of a vector space (lets say $K^n$ where $K$ are the rationals).
Here is how to obtain all exterior powers for $n=3$ using GAP:
n:=3;
V:=Rationals^n;
Display(Basis(V));
n:=Dimension(V);
U:=[];for i in [0..n] do Append(U,[ExteriorPower(V,i)]);od;U;
Question 1: Is there an easy way how to obtain the exterior algebra of $K^n$ in GAP as an algebra over $K$ with identity? That is, can one also obtain the multiplication for two elements in the exterior powers using GAP?
I can do question 1 using the GAP-package QPA via quivers, but then one loses the ability to use (some) linear algebra commands from GAP.
Question 2: Now let $M: K^n-> K^n$ be a matrix, then we can also use the command Mi:=ExteriorPower(M,i) in GAP to get the matrix for the extension $M^i$ of $M$ to the $i$-th exterior power of $K^n$. Is there a direct way to get an explicit basis of the kernel of Mi in GAP (so that the elements of the kernel are really elements of ExteriorPower(K^n,i) and not just "normal" vectors with entries in $K$)?
Of course I know how to translate the output of GAP into what I want for question 2, but I feel like there should be a direct command for that that I am not aware of.
Thanks for any help