I'm currently going through a chapter of Arnolds Mathematical Methods of Classical Mechanics and got stuck in Chapter 33 Exterior Multiplication.
Define the exterior product of a k-form $\omega^k$ on $\mathbb{R}^n$ and an l-form $\omega^l$ on $\mathbb{R}^n$ as $(\omega^k \wedge \omega^l)(a_1,\dots,a_{k+l}) = \sum (-1)^\nu \omega^k(a_{i_{1}},\dots,a_{i_{k}}) \omega^l(a_{j_{1}},\dots,a_{j_{l}})$, where the sum goes over all permutations (indexed by the $(i_1,\dots ,i_k,j_1,\dots ,j_l)$) of $(1,\dots,k+l)$ and $\nu$ takes on the value 1 for an even resp. -1 for an odd number of permutations overall. I want to check, whether $(\omega^k \wedge \omega^l) (a_1,\dots, a_r ,\dots, a_s, \dots,a_{k+l}) = -(\omega^k \wedge \omega^l)(a_1,\dots, a_s ,\dots, a_r, \dots,a_{k+l})$, but I'm a little confused about the notation. Aren't all possible permutations already in the sum, so that there shouldn't be a skew-symmetrical dependence on the input vectors?