  Since the present $\omega$ meson family has not been established, in this
work, we carry out an investigation of the mass spectrum and Okubo-Zweig-Iizuka
a allowed two-body strong decay of the $S$-wave and the $D$-wave $\omega$
mesons, and make the comparison with the experimental data of these reported
$\omega$ states and the $\omega$-like $X(2240)$ state observed by BESIII. By
this study, we not only suggest the possible assignments to these observed
$\omega$ states under the framework of the $\omega$ meson family, but also
predict three $\omega$ mesons ($\omega(5S)$, $\omega(2D)$ , and $\omega(4D)$)
which are still missing in experiment. The present study may provide valuable
information to further construct the $\omega$ meson family. Considering the
present running status of BESIII, we also suggest that BESIII should pay more
attention to the issue of $\omega$ meson with accumulating more data.
