Symmetric Relation question

I know that a symmetric relation on a set X with for every a,b ∈X is noted by:

aRb ⇒ bRa

But I was wondering if this could be an alternative definition (please explain why it is not if this fails, thank you).

A relation on set X is symmetric iff for every (x,y) ∈X, (x, y) ∈X ⇒(y, x) ∈X

The reason I suspect this might be true is because the "relation" might be defined as an ordered pair. x is related to y by x and y matched as an ordered pair, (x, y).

Thanks for your time.