Odd Spheres $S^{2n-1}$ are Sasaki-Einstein manifolds. On the Wikipedia page one can find a mention about how we can see it explicitly as an embedding in $\mathbb{C}^n$. However I would like to construct all the relevant forms without using this embedding. How can I do?
In particular, the Reeb vector $\xi$, the contact form $\sigma$ (its dual) and the two forms (real and complex respectively) $\omega$ and $\Omega$ satisfying the following
$$ \xi \lrcorner \omega = \xi \lrcorner \Omega = 0$$
$$ \Omega \wedge \omega = \Omega \wedge \Omega =0 $$ $$\mathrm{vol}_{S^{2n-1}} = \sigma \wedge \omega \wedge \omega = \tfrac{1}{2} \sigma \wedge \Omega \wedge \bar{\Omega}$$
and some differential conditions
$$\mathrm{d}\sigma = 2 \omega$$ $$\mathrm{d}\Omega = 3 i \sigma \wedge \Omega$$ and for the metric
$$\mathrm{d}s^2 = \sigma \otimes \sigma + \mathrm{d}s^2(B_{KE})$$
where $B_{KE}$ is the Kähler-Einstein base.