Is there a relation which is neither symmetric nor antisymmetric?

If every pair satisfies $aRb\rightarrow bRa$ then the relation is symmetric. If there is at least one pair which fails to satisfy that then it is not symmetric.

Similarly if there is at least one pair which has $(aRb\rightarrow bRa)\land a\neq b$ then antisymmetry is also not satisfied.

We can therefore take the following relation: $\{a,b,c\}$ would be our universe and $R=\{\langle a,b\rangle,\langle b,a\rangle,\langle a,c\rangle\}$.

The fact that $aRc\land\lnot cRa$ shows that the relation is not symmetric, but $a\neq b$ and both $aRb$ and $bRa$ hold.