How do I prove that homomorphism $\phi : \; Mod(S_g)\to Sp(2g, \mathbb{Z})$ (induced by the action of mapping class group of a surface on integer homologies of a surface) is an epimorphism? My idea was to work with generators, but I was not able to prove it this way.

I would love to get detailed answers in order to understand this better. Thanks in advance.