I'm trying to compute $\langle dx^\mu\wedge dx^\nu, dx^\rho\wedge dx^\sigma \rangle$. This should give give the answer $G^{\mu\rho}G^{\nu\sigma}-G^{\mu\sigma}G^{\nu\rho}$, if we use the formula $$\langle v_1\wedge v_2,w_1\wedge w_2 \rangle=\det(\langle v_i, w_j\rangle)$$
However, if this is the case, I wouldn't get the Yang mill's action $$\int_M tr(F\wedge *F)=\int_M \text{tr}(F_{\mu\nu}F_{\rho\sigma})G^{\mu\rho}G^{\nu\sigma}\text{vol}$$ (* is the Hodge star operator), and should get instead
$$\int_M \text{tr}(F_{\mu\nu}F_{\rho\sigma})(G^{\mu\rho}G^{\nu\sigma}-G^{\mu\sigma}G^{\nu\rho})\text{vol}$$
Anyone knows where I commit a mistake?