Show that $i^*\omega \in \Omega^{n}(S^n)$ is an orientation form on $S^n$. Here $i:S^n \to \mathbb{R}^{n+1} - \{0\}$ and $\omega$ is a $n$-form on $\mathbb{R}^{n+1}- \{0\}$ for which $\omega_p(v_1, \dots, v_n) = \det(p,v_1, \dots, v_n).$
So to prove that this is an orientation form one needs to show that $(i^*\omega)_p \ne 0$ for every $p \in S^n$. Let $p \in S^n$ and consider $v_1, \dots,v_n \in T_pS^n$. We have that $$\begin{align*}(i^*\omega)_p(v_1, \dots, v_n) &= \omega_{i(p)}(di_p(v_1), \dots, di_p(v_n)) \\ &= \det(i(p), di_p(v_1), \dots, di_p(v_n)).\end{align*}$$
Now to show that this determinant is non-zero is where I'm getting stuck at. What options do I have here to show that this is either negative or positive?