I need to prove $\omega_{\mu\nu}$ it's a antisymmetric: $$\omega_{\mu\nu}=-\omega_{\nu\mu}$$ according to these property: \begin{align*}g_{\rho\sigma} = g_{\mu\nu}\Lambda^\mu_{\ \ \ \rho} \Lambda^\nu_{\ \ \ \sigma} \end{align*}
$$\Lambda^\mu_{\ \ \ \nu}=\delta^\mu_{\ \ \ \nu} + \omega^\mu_{{\ \ \ \nu}}$$ then:
$$(\delta_{\rho}^{\mu}+\omega_{\rho}^{\mu})g_{\mu\nu}(\delta_{\sigma}^{\nu}+\omega_{\sigma}^{\nu})+o(\omega^{2})=g_{\rho\sigma}$$
We get this:
$$\tag{*}g_{\rho\sigma}+\omega_{\rho}^{\mu}g_{\mu\nu}\delta_{\sigma}^{\nu}+\omega_{\sigma}^{\nu}g_{\mu\nu}\delta_{\rho}^{\mu}+o(\omega^2)=g_{\rho\sigma}$$ Finally: $$g_{\rho\sigma}+\omega_{\rho\sigma}+\omega_{\sigma\rho}=g_{\rho\sigma}$$
just move terms we can get the correct answer right? But I have no idea about the third term of equation $(*)$: $$\omega_{\sigma}^{\nu}g_{\mu\nu}\delta_{\rho}^{\mu}$$ I want to know why not in this order: $$\delta_{\rho}^{\mu}g_{\mu\nu}\omega_{\sigma}^{\nu}$$ then we get this wrong result: $$g_{\rho\sigma}+\omega_{\rho\sigma}+\omega_{\rho\sigma}=g_{\rho\sigma}$$
Can someone please just told me how to get the correct order?