Symmetric, skew-symmetric $\mathbb{R}$-bilinear form, matrix of form in $\mathbb{R}$-basis.

Question

Let $(-, -): V \times V \to \mathbb{C}$ be a positive definite Hermitian inner product on a finite dimensional $\mathbb{C}$-vector space $V$ and let $e_1, \ldots, e_n$ be an orthonormal basis of $V$. For any $v$, $w \in V$, let $(v, w)_{\text{Re}}$ be the real, resp. $(v, w)_{\text{Im}}$ the imaginary, part of the complex number $(v, w)$. Thus, we have$$(v, w) = (v, w)_{\text{Re}} + i \cdot (v, w)_{\text{Im}}.$$Let $V_\mathbb{R}$ be $V$ viewed as a vector space over $\mathbb{R}$, so $e_1, \ldots, e_n, i \cdot e_1, \ldots, i \cdot e_n$, is an $\mathbb{R}$-basis of $V_\mathbb{R}$.

1. How do I see that $v$, $w \mapsto (v, w)_\text{Re}$, resp. $v$, $w \mapsto (v, w)_{\text{Im}}$, is a symmetric, resp. skew-symmetric, $\mathbb{R}$-bilinear form on $V_\mathbb{R}$?
2. What is the matrix of that bilinear form in the above $\mathbb{R}$-basis?

Answer 1

How do I see that $v$, $w \mapsto (v, w)_\text{Re}$, resp. $v$, $w \mapsto (v, w)_{\text{Im}}$, is a symmetric, resp. skew-symmetric, $\mathbb{R}$-bilinear form on $V_\mathbb{R}$?

Since $$(v,w)=\overline{(w,v)}=\overline{(w,v)_\mathrm{Re}+i(w,v)_\mathrm{Im}}=(w,v)_\mathrm{Re}-i(w,v)_\mathrm{Im},$$ we have $$(v,w)_\mathrm{Re}+i(v,w)_\mathrm{Im}=(w,v)_\mathrm{Re}-i(w,v)_\mathrm{Im}$$ that is $$(v,w)_\mathrm{Re}=(w,v)_\mathrm{Re},\quad (v,w)_\mathrm{Im}=-(w,v)_\mathrm{Im}.$$

What is the matrix of that bilinear form in the above $\mathbb{R}$-basis?

Find it directly. What is in the $(j,k)$ cell of the matrix?