First by writing out the terms and re-arranging the subscripts and using the skew symmetry of $A_{i\alpha\beta}$ we get

(3.116)				$\displaystyle A_{i,\alpha+1,\beta}A_{j,\alpha+2,\gamma}y_{\beta}y_{\gamma}dy_{\alpha}$
			$\displaystyle=$	$\displaystyle\Big(A_{i,\alpha+1,\alpha}A_{j,\alpha+2,\alpha}y_{\alpha}^{2}+A_{i,\alpha+1,\alpha}A_{j,\alpha+2,\alpha+1}y_{\alpha}y_{\alpha+1}$
				$\displaystyle+A_{i,\alpha+1,\alpha+2}A_{j,\alpha+2,\alpha}y_{\alpha}y_{\alpha+2}+A_{i,\alpha+1,\alpha+2}A_{j,\alpha+2,\alpha+1}y_{\alpha+1}y_{\alpha+2}\Big)dy_{\alpha}$
			$\displaystyle=$	$\displaystyle A_{i,\alpha,\alpha+1}A_{j,\alpha,\alpha+2}y_{\alpha}^{2}dy_{\alpha}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1}y_{\alpha}y_{\alpha+2}dy_{\alpha+2}$
				$\displaystyle-A_{i,\alpha,\alpha+2}A_{j,\alpha,\alpha+1}y_{\alpha}y_{\alpha+1}dy_{\alpha+1}-A_{i,\alpha,\alpha+1}A_{j,\alpha,\alpha+1}y_{\alpha}y_{\alpha+1}dy_{\alpha+2}$

Now we can skew-symmetrize with respect to $i$ and $j$

(3.117)

$$A_{i,\alpha+1,\beta}A_{j,\alpha+2,\gamma}y_{\beta}y_{\gamma}dy_{\alpha}\wedge dx_{i}\wedge dx_{j}=\frac{1}{2}(A_{i,\alpha+1,\beta}A_{j,\alpha+2,\gamma}-A_{j,\alpha+1,\beta}A_{i,\alpha+2,\gamma})y_{\beta}y_{\gamma}dy_{\alpha}\wedge dx_{i}\wedge dx_{j}$$

Correspondingly by skew-symmetrizing each term of (3.116) with respect to $i$ and $j$, we get

(3.118)				$\displaystyle\frac{1}{2}(A_{i,\alpha+1,\beta}A_{j,\alpha+2,\gamma}-A_{j,\alpha+1,\beta}A_{i,\alpha+2,\gamma})y_{\beta}y_{\gamma}dy_{\alpha}$
			$\displaystyle=$	$\displaystyle\frac{1}{2}(A_{i,\alpha,\alpha+1}A_{j,\alpha,\alpha+2}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1})y_{\alpha}(y_{\alpha}dy_{\alpha}+y_{\alpha+1}dy_{\alpha+1}+y_{\alpha+2}dy_{\alpha+2})$
			$\displaystyle=$	$\displaystyle\frac{1}{2}(A_{i\alpha,\alpha+1}A_{j\alpha,\alpha+2}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1})y_{\alpha}\cdot rdr$

∎