An exterior algebra is a polynomial ring where multiplication is mildly noncommutative, in that, for every x and y in the ring, y*x = (1)^(deg(x) deg(y)) x*y, and that for every x of odd degree, x*x = 0.In Macaulay2, deg(x) is the degree of x, or the first degree of x, in case a multigraded ring is being used. The default degree for each variable is 1, so in this case, y*x = x*y, if x and y are variables in the ring.
Create an exterior algebra with explicit generators by creating a polynomial ring with the option SkewCommutative.
i1 : R = QQ[x,y,z, SkewCommutative => true]
o1 = R
o1 : PolynomialRing, 3 skew commutative variables

i2 : y*x
o2 = x*y
o2 : R

i3 : (x+y+z)^2
o3 = 0
o3 : R

i4 : basis R
o4 =  1 x xy xyz xz y yz z 
1 8
o4 : Matrix R < R

i5 : basis(2,R)
o5 =  xy xz yz 
1 3
o5 : Matrix R < R

i6 : S = QQ[a,b,r,s,t, SkewCommutative=>true, Degrees=>{2,2,1,1,1}];

i7 : r*a == a*r
o7 = false

i8 : a*a
o8 = 0
o8 : S

i9 : f = a*r+b*s; f^2
o10 = 2a*b*r*s
o10 : S

i11 : basis(2,S)
o11 =  a b rs rt st 
1 5
o11 : Matrix S < S

All modules over exterior algebras are right modules. This means that matrices multiply from the opposite side:
i12 : x*y
o12 = x*y
o12 : R

i13 : matrix{{x}} * matrix{{y}}
o13 =  xy 
1 1
o13 : Matrix R < R

You may compute GrÃ¶bner bases, syzygies, and form quotient rings of these skew commutative rings.