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If I understand correctly, the wedge product a^b of 2 vectors a, b is
synonym of their outer product as defined in geometric algebra (see,
e.g., Fig. 2 in [1]).
The fact that by-vector a^b lives in (a, b)-plane does not eliminate handedness.
Just as the line perpendicular to (a, b) may have two opposite directions,
the parallelogram representing by-vector a^b is a 2-sided plane segment.
Calling the opposite sides "black" and "white" respectively, we have to decide which one of them will face,
Reference [1]: D. Hestenes, Am. J. Phys., 71 (2), 2003