You can make explicit an homotopy showing that the map $(\chi,\xi)\mapsto \chi\smile\xi\pm\xi\smile\chi$ is homotopic to zero. Using the explicit homotopy, you can generalize your formula for $\chi\smile\chi$.

You can make explicit an homotopy showing that the map $(\chi,\xi)\mapsto \chi\smile\xi\pm\xi\smile\chi$ is homotopic to zero. Using it, you can generalize your formula for $\chi\smile\chi$.