Is the wedge product of top forms zero?

Question

1. Let's take $\omega, \tau \in \Omega^n\mathbb R^n$. I think $\omega=fdx^1 \wedge ... \wedge dx^n$ and $\tau=gdx^1 \wedge ... \wedge dx^n$ for some $f,g \in C^{\infty}(\mathbb R^n)$. Then $$\omega \wedge \tau = fg dx^1 \wedge ... \wedge dx^n \wedge dx^1 \wedge ... \wedge dx^n$$ Is $dx^1 \wedge ... \wedge dx^n \wedge dx^1 \wedge ... \wedge dx^n$ automatically zero because of share indices?

We can generalise to manifolds, but let's concentrate on $\mathbb R^n$ for now.

2. I think we can generalise by pigeonhole principle to say for $\mathbb R^n$ if $\omega$ is a $k$-form and $\tau$ is an $l$-form and $k+l > \frac n 2$, then their wedge product must be zero. Is that correct?

Thanks in advance!