Marks (J. Algebra 280 (2004) 463–471) proved that if the skew polynomial ring $R[x; \sigma]$ is left or right duo, then $R[x; \sigma]$ is commutative. It is proved that if $R[x; \sigma]$ is weakly left (resp., right) duo over a reduced ring $R$ with an endomorphism (resp., a monomorphism) $\sigma$, then $R[x; \sigma]$ is commutative. This concludes that a noncommutative skew polynomial ring is not weakly left duo when the base ring is reduced. It is also shown that if $R[x; \sigma]$ is weakly left duo then the polynomial ring $R[x]$ is weakly left duo. We next study the structure of the Ore extension $R[x; \sigma, \delta]$ when it is weakly left or right duo.