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I suppose that the question wants to build the corresponding matrix in the peculiar order of all monomials of degree $\le 2$ given above, that works for all (not so many) possible polynomials of degree $\le 2$ in the given ring. The hint MacCaulay was actively ignored in my following answer / sample code, since i considered the task as a task of identifying coefficients of polynomials.

F = GF(2)
R.<x1,x2,x3> = PolynomialRing( F )
p, q = x1*x2 + x1*x3 + x1, x1 + x2 + 1

degrees = ( (2,0,0),
            (1,1,0),
            (0,2,0),
            (1,0,1),
            (0,1,1),
            (0,0,2),
            (1,0,0),
            (0,1,0),
            (0,0,1),
            (0,0,0), )
d = len(degrees)

print matrix( F, 2, d, [ [ pol.coefficient( dict( zip( (x1,x2,x3) , degtuple ) ) )
                                            for degtuple in degrees ]
                         for pol in ( p, q ) ] )

This gives:

[0 1 0 1 0 0 1 0 0 0]
[0 0 0 0 0 0 1 1 0 1]