# Why antisymmetric construction(wedge product) was so special in tensor compare to symmetric construction?

Gravitation By Charles Misner, Kip Thorene, John Wheeler page 83. The book introduced the several ways of constructing new tensors from the old.

A pair of such process was by symmetrization and antisymmetrization. However,in the immediate part, the book noted that wedge product (especially for rank 2 cases, i.e. bivector, $$u\wedge v$$, 2-form, $$\omega\wedge \mu$$) was antisymmetrized.

Though in higher dimensional cases, such as Trivector(i.e. $$u\wedge v\wedge w$$) it becomes more complected, it seemed to emphasize that there's something very special about the antisymmetric process. A related post I found was here: Relationship between tensor product and wedge product which indicated the correspondence between antisymmetrization to the quotient group as expected.

With the reason above(it's relation to quotient group), and the fact that there's process to convert antisymmetric tensor to symmetric tensor, the utilization of antisymmetric process was therefore somewhat evidentially preferred.

However, I'm still having some doubts.

1. Is there any other reason for us to utilize wedge product in such favorable fashion?

2. Since there's a direct connection between symmetrization and antisymmetrization, is there a counter part of antisymmetric process(wedge product), i.e. symmetric process? If there is, how does it look like and how can we use it? (i.e. there seemed to be some related considerations in second and third rank duals, and the linear dependence, etc. but I'm not very convinced if that's it, since it's still far from what we could achieve with wedge product.) If not, is there any argument to indicate the disadvantage for the symmetric process?