Let's suppose one has a $\mathbb{R}$ vector space $V$ such that $\{v_1,\ldots,v_n\}$ is an orthogonal basis. Let one call $\alpha_i:V\to \mathbb{R}$ the orthogonal projection of a vector on $v_i$. It is, $\alpha_i(w) = \left<w,v_i\right>$.
Then, one easily sees that $\{\alpha_1,\ldots,\alpha_n\}$ form a basis of $V^*$. Now, let one consider $d\alpha_1$. It is known that one may write:
$$ d\alpha_1 = \sum_{i<j} a_{ij}\cdot \alpha_i \wedge \alpha_j.$$
I can't manage to say anything about these coefficients $a_{ij}$. In fact, what my main goal is understanding the dimension of $\ker(d\alpha_1)^n$ (where the exponent stands for the wedge product of $d\alpha_1$ done $n$ times).
So you're switching from linear algebra to thinking of $V$ as a manifold, of course. If we introduce (global) coordinates on $V$ by writing $x=\sum x_iv_i$, then by definition $\alpha_i = dx_i$ and so $d\alpha_i = 0$.