Interior product on multivectors and antisymmetry

Question

Let $V$ be a vector space, and define the interior product of $\sigma \in V^*$ with a multivector $v_1\wedge \cdots \wedge v_n$ by : $$ int(\sigma)(v_1\wedge \cdots \wedge v_n) = \sum_{i} (-1)^{i+1} \sigma(v_i) v_1\wedge \cdots \widehat v_i \cdots \wedge v_n $$ I want to prove that $int(\sigma)int(\tau)=-int(\tau)int(\sigma)$ for all $\sigma, \tau\in V^*$, but I don't know how to proceed. Applying the formula twice doesn't seem to yield the result in an obvious way (to me).