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I'm pretty soft on this subject but one thing that's always bugged me when I see bivectors represented as a parallelograms is that different coplanar parallelograms of the same area are the same bivector algebraically but visually very different. It just feels confusing when you're trying to learn what it "is".
Isn't that just inherent in binary operations? I feel like you are saying the same thing as: What is 0 sometimes people write 1-1 or 3-3 or -10+10.
by the linearity of ^, B = a^b = 1 * a^b = x*(x^-1) a^b = xa ^ (x^-1)b
So the bivector is the "hyperbola" of all touching parallelograms just like 0 is the set of all oppositely directed arrows of equal magnitude.
But the parallelograms only become a bivector after you wedge them. not before.