I've already made a post about this topic here, but I realized that I didn't understand the explanation on that post. in Chapter 7 of Rindler's book on relativity, in section about electromagnetic field tensor, he states that

_and introducing a factor 1/c for later convenience, we can ‘guess’ the tensor equation_, $$ F_\mu= \frac{q}{c} E_{\mu \nu} U^\
\nu$$
_thereby introducing the electromagnetic field tensor_$$E_{\mu \nu}$$
_We would surely want the
force $F\mu$ to be rest-mass preserving, which, according to (6.44) and (7.15), requires_
$$F\mu U^\mu = 0$$. _So we need_
$$E_{\mu \nu} U^\mu U^\nu = 0$$ 
_for all $ U^\mu$ , and hence the antisymmetry of the field tensor_
$$E_{\mu \nu}= −E_{\nu \mu}$$\\

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 I'm really confused about the correct way to show that the equation $E_{\mu \nu} U^\mu U^\nu = 0$ implies the fact that $E_{\mu\nu}$ is antisymmetric tensor. What is  the correct demonstration of this implication? 

OBS: i've e saw some posts answering this kind of question with bilinear maps notation, instead of component notation. If possible, please make some demonstration using the index notation as in the post.