When we have a symplectic form $\omega$ on an even dimensional linear space, we can consider the complex structure $J$ that is compactible with it, i.e. $\omega(Jv,Jw)=\omega(v,w)$.
It is always possible to find such structure locally, i.e. in linear spaces. In Darboux chart, the symplectic form is always the standard form, in this case, we can just take the standard complex structure $J_0$.
My question is, for an arbitary symplectic manifold, is it alwyas possible to find such global compactible $J$? If possible, how do we construct such $J$ explicitly?
Yes it is always possible to put a compatible almost complex structure on a symplectic manifold. This is due to Gromov I believe, any text book on symplectic geometry will have this (I like "Introduction to Symplectic Topology" by Mcduff and Salamon). The proof is quite topological so the almost complex structures are usually non-explicit.
Although there are exceptions, for example the compatible almost complex structure on $S^2$ (which happens to be integrable) may be described by taking cross product with an outward pointing normal vector of unit length. There are other "nice situations" but in general the compatible almost complex structure will be quite non-explicit.