I am trying to understand the proof of the Kähler identities on a Kähler manifold, and as a first step want to understand the proof on $\mathbb{C}^n$, as presented in https://sites.math.washington.edu/~lee/Courses/549-2004/proof.pdf.
It says there: "We begin by working on $\mathbb{C}^n$ with Euclidean metric $\overline{g}$." Later the inner product $\langle dz^j, dz^k \rangle$ on 1-forms is used.
Considering the coordinates $(x_1, ix_1, \dots, x_n, ix_n)$ on $\mathbb{C}^n$, we another set of coordinates $(z_1, \overline{z_1}, \dots, z_n, \overline{z_n})$ given by $z_j=x_j-ix_j$ and $\overline{z_j}=x_j-ix_j$.
Question: How is the aforementioned Euclidean metric given in these coordinates? How is the Hermitian metric $\langle , \rangle$ from the reference given in these coordinates? Why is the formula $\langle dz^j, dz^{\overline{k}} \rangle = \overline{g} \left( dz^j, dz^k \right)$ true?
My thoughts on this: The formulae $\overline{g} \left( dz^j, dz^k \right)=0$ and $\overline{g} \left( dz^j, dz^{\overline{k}} \right)$ suggest that $$g=\begin{pmatrix} 0 & 1 & & 0&0 \\ 1 & 0 & \dots & 0&0 \\ & \vdots & &\vdots& \\ 0&0& \dots & 0&1\\ 0&0& & 1&0\end{pmatrix}$$ written in coordinates $(z_1,\overline{z_1},\dots,z_n,\overline{z_n})$. However, this is not Euclidean. What is actually meant?