algebra - How can I implement anti-commuting (Grassmann) numbers (variables)?

I am looking for a way to define anti-commuting numbers (variables) in Mathematica. Let say we have a set of four variables like ${x_1,x_2,x_3,x_4}$ from which I need to get all possible permutations, then multiply them so they appear as a sum as

$\qquad x_1*x_2*x_3*x_4+x_1*x_2*x_4*x_3+(22\,{\rm terms})$

But these variables are Grassmann anti-commutative, so whenever we interchange any two of them we should get a negative sign, i.e., $x_1*x_2 = -x_2*x_1$, for example the above expression should appear as

$\qquad x_1*x_2*x_3*x_4-x_1*x_2*x_4*x_3+(22\,{\rm terms})$

I know how to permute the variables, but I do not know how get their signs correctly. Any ideas?

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