Let $(M,\omega)$ is a symplectic manifold, and you can assume it is compact if necessary. Denote by $\mathcal J^k = \mathcal J^k(M, \omega)$ the set of all $\omega$-compatible almost complex structures of class $C^k$.

How can we see that the tangent space $T_J\mathcal J^k$ at $J$ consists of $C^k$-sections $Y$ of the bundle $End(TM,J,\omega)$ whose fiber at $x\in M$ is the space of linear maps $Y:T_xM\to T_xM$ such that $$JY+YJ=0,~~~~~~~~\omega(Yv,w)+\omega(v,Yw)=0$$ ? One intuitive way is to consider a "curve" $J_t=J+tY+o(t)$, and plug in this to $J_t^2=1$ and $J_t^*\omega(u,v)=\omega(u,v)$.

However, I prefer a rigorous way using explicit local coordinates of the space $\mathcal J^k$. Please help, thanks!