I can't understand the following basic thing:
Let $X$ be a complex manifold of dimension $n$ and $T_xX$ its tangent space in $x\in X$. Then on $T_xX$ we can define a hermitian form $\sum_{i=1}^ndz_id\overline{z}_i$. I know that $\omega:=-\mathcal{Im}h$ is an alternating form and that it can be written as $\frac i 2 \sum_{i=1}^ndz_i\wedge d\overline{z}_i$.
My question is: Why $\omega$ can be written like $\frac i 2 \sum_{i=1}^ndz_i\wedge d\overline{z}_i$?
Which is equivalent to asking why is it $-\mathcal{Im}dz_id\overline{z}_i=dz_i\wedge d\overline{z}_i$?
If I compute $dz_id\overline{z}_i$ I get $-\mathcal{Im}dz_id\overline{z}_i=dx_idy_i-dy_idx_i=0$... Where am I wrong?
Of course this is just a problem of linear algebra, but I formulated it in the complex geometry context to maintain the notation
You're wrong to say that $dx\,dy-dy\,dx=0$. You need to put tensor products into everything and then the definition of wedge product is precisely $dx\wedge dy = dx\otimes dy-dy\otimes dx$. (Some people will put a factor of $1/2$ there, but I don't.)