  Recent pulsar timing array collaborations have reported evidence of the
stochastic gravitational wave background. The gravitational waves induced by
primordial curvature perturbations, referred to as scalar-induced gravitational
waves (SIGWs), could potentially be the physical origins of this gravitational
wave background. In addition to the statistical properties of SIGWs derived
from the primordial fluctuations, SIGWs also have intrinsic non-Gaussianity
originating from the non-linear interactions of Einstein's gravity, which,
however, is rarely explored. In this paper, we study the intrinsic
non-Gaussianity of SIGWs with the bispectrum and the skewness under the
assumption of the Gaussian primordial curvature perturbations,
phenomenologically modeled as a lognormal spectrum. The bispectrum is shown to
be vanishing in the collinear limit, which is independent of the initial
conditions and the dynamics of SIGWs. For the SIGWs generated during the
radiation-dominated era, the bispectrum is of flatten-type non-Gaussianity,
with only four polarization components left to be non-vanishing. Additionally,
in order to obtain the correct bispectrum, we also propose a time-oscillation
average scheme and a regularization scheme. Utilizing the skewness for
quantifying the degree of non-Gaussianity, it is found that the curvature power
spectrum with a narrow width can result in an enhancement of the third-order
non-Gaussianity. The conclusion holds for both the SIGWs generated in the
radiation-dominated era and the matter-dominated era.
