  Extracting the non-Gaussian information of the cosmic large-scale structure
(LSS) is vital in unlocking the full potential of the rich datasets from the
upcoming stage-IV galaxy surveys. Galaxy skew spectra serve as efficient
beyond-two-point statistics, encapsulating essential bispectrum information
with computational efficiency akin to power spectrum analysis. This paper
presents the first cosmological constraints from analyzing the full set of
redshift-space galaxy skew spectra of the data from the SDSS-III BOSS,
accessing cosmological information down to nonlinear scales. Employing the
${\rm S{\scriptsize IM}BIG}$ forward modeling framework and simulation-based
inference via normalizing flows, we analyze the CMASS-SGC sub-sample, which
constitute approximately 10\% of the full BOSS data. Analyzing the scales up to
$k_{\rm max}=0.5 \, {\rm Mpc}^{-1}h$, we find that the skew spectra improve the
constraints on $\Omega_{\rm m}, \Omega_{\rm b}, h$, and $n_s$ by 34\%, 35\%,
18\%, 10\%, respectively, compared to constraints from previous ${\rm
S{\scriptsize IM}BIG}$ power spectrum multipoles analysis, yielding
$\Omega_{\rm m}=0.288^{+0.024}_{-0.034}$, $\Omega_{\rm b}=
0.043^{+0.005}_{-0.007}$, $h=0.759^{+0.104}_{-0.050}$, $n_{\rm s} =
0.918^{+0.041}_{-0.090}$ (at 68\% confidence limit). On the other hand, the
constraints on $\sigma_8$ are weaker than from the power spectrum. Including
the Big Bang Nucleosynthesis (BBN) prior on baryon density reduces the
uncertainty on the Hubble parameter further, achieving
$h=0.750^{+0.034}_{-0.032}$, which is a 38\% improvement over the constraint
from the power spectrum with the same prior. Compared to the ${\rm
S{\scriptsize IM}BIG}$ bispectrum (monopole) analysis, skew spectra offer
comparable constraints on larger scales ($k_{\rm max}<0.3\, {\rm Mpc}^{-1}h$)
for most parameters except for $\sigma_8$.
