A Poisson structure on a manifold $M$ is defined to be a binary operation $\{-,-\}:C^\infty(M)\times C^\infty(M)\rightarrow C^\infty(M)$ satisfying certain conditions.
Other ways to see a Poisson structure is as a bisector field on $M$ (as an element of $\mathfrak{X}^2(M))$. But, I am not able to find a reference that gives details about a one-one correspondence between these two ways.
Let $\{-,-\}:C^\infty(M)\times C^\infty(M)\rightarrow C^\infty(M)$ be a Poisson structure on $M$. Fix $f\in C^\infty(M)$. This gives a map $\{f,-\}:C^\infty(M)\rightarrow C^\infty(M)$ which turns out to be a derivation (thanks to the property of the binary operation $\{-,-\}$). So, we get a vector field $\{f,-\}$ which is usually denoted by $X_f$.
This procedure gives the map $C^\infty(M)\rightarrow \mathfrak{X}(M)$ defined as $f\mapsto \{f,-\}$ for $f\in C^\infty(M)$. This would induce the map $\Omega^1(M)\rightarrow \mathfrak{X}(M)$ (locally) defined as $df\mapsto \{f,-\}$. Seeing $\Omega^1(M)$ as $\Gamma(M,T^*M)$ and $\mathfrak{X}(M)$ as $\Gamma(M,TM)$ gives us a map of sections $\Gamma(M,T^*M)\rightarrow \Gamma(M,TM)$.
Not all maps of sections need to come from a map of vector bundles, but, this being a $C^\infty(M)$-linear map of sections, comes from a map of vector bundles $T^*M\rightarrow TM$.
Any map $\Phi:V^*\rightarrow V$ can be seen as a map $\Phi:V^*\times V^*\rightarrow \mathbb{R}$ by declaring $\Phi(\theta, \psi)=\psi(\Phi(\theta))$ for $(\theta,\psi)\in V^*\times V^*$.
In same fashion, the map $T^*M\rightarrow TM$ can be seen as a map $T^*M\times T^*M\rightarrow M\times \mathbb{R}$ which can in turn be seen as a map $\Omega^1(M)\times \Omega^1(M)\rightarrow C^\infty(M)$ which is the notion of a bisector filed.
Is there any reference (book/lecture notes) that gives the above details (and little more than that)?