  Extracting the non-Gaussian information encoded in the higher-order
clustering statistics of the large-scale structure is key to fully realizing
the potential of upcoming galaxy surveys. We investigate the information
content of the redshift-space {\it weighted skew spectra} of biased tracers as
efficient estimators for 3-point clustering statistics. The skew spectra are
constructed by correlating the observed galaxy field with an
appropriately-weighted square of it. We perform numerical Fisher forecasts
using two synthetic datasets; the halo catalogs from the Quijote N-body
simulations and the galaxy catalogs from the Molino suite. The latter serves to
understand the effect of marginalization over a more complex matter-tracer
biasing relation. Compared to the power spectrum multipoles, we show that the
skew spectra substantially improve the constraints on six parameters of the
$\nu\Lambda$CDM model, $\{\Omega_m, \Omega_b, h, n_s, \sigma_8, M_\nu\}$.
Imposing a small-scale cutoff of $k_{\rm max}=0.25 \, {\rm Mpc}^{-1}h$, the
improvements from skew spectra alone range from 23% to 62% for the Quijote
halos and from 32% to 71% for the Molino galaxies. Compared to the previous
analysis of the bispectrum monopole on the same data and using the same range
of scales, the skew spectra of Quijote halos provide competitive constraints.
Conversely, the skew spectra outperform the bispectrum monopole for all
cosmological parameters for the Molino catalogs. This may result from
additional anisotropic information, particularly enhanced in the Molino sample,
that is captured by the skew spectra but not by the bispectrum monopole. Our
stability analysis of the numerical derivatives shows comparable convergence
rates for the power spectrum and the skew spectra, indicating potential
underestimation of parameter uncertainties by at most 30%.
