  Using 2.92 fb$^{-1}$ of electron-positron annihilation data collected at a
center-of-mass energy of $\sqrt{s}= 3.773$ GeV with the BESIII detector, we
present an improved measurement of the branching fraction $\mathcal{B}(D^+ \to
\omega e^+ \nu_{e}) = (1.63\pm0.11\pm0.08)\times 10^{-3}$. The parameters
defining the corresponding hadronic form factor ratios at zero momentum
transfer are determined for the first time, we measure them to be $r_V =
1.24\pm0.09\pm0.06$ and $r_2 = 1.06\pm0.15 \pm 0.05$. The first and second
uncertainties are statistical and systematic, respectively. We also search for
the decay $D^+ \to \phi e^+ \nu_{e}$. An improved upper limit $\mathcal{B}(D^+
\to \phi e^+ \nu_{e}) < 1.3 \times 10^{-5}$ is set at 90\% confidence level.
