Let $X$ be a Riemann surface ,we call $\phi:X\to \Bbb{P}^n$ holomorphic at $p\in X$ iff there exist holomorphic function $g_0,...,g_n$ defined on $X$ near $p$ not all zero at $p$ such that $\phi(x) = [g_0(x):...:g_n(x)]$.
I am not sure if this definition is equivalent to the standard definition of a holomorphic map (between complex manifolds).
One direction is clear that is the holomorphic map defined in the first line is a holomorphic map in the standard definition.
Conversely is it true that all the holomorphic maps in the standard definition, always exist a set of holomorphic $g_i$ locally representing this map?
That's not hard.For the converse direction assume $\phi:X\to \Bbb{P}^n$ is holomorphic map between complex manifold .
Since $\phi:X\to \Bbb{P}^n$ maps into $\Bbb{P}^n$ therefore $\phi(x) = [\phi_0(x):...:\phi_n(x)]$ is not all zero assume $\phi_0(x)\ne 0$ ,let $U_0 = \{z_0\ne 0\}$ then $\phi(x) = [1:\tilde{\phi_1}(x):...:\tilde{\phi_n}(x)]$ on $\phi^{-1}(U_0)\to U_0$, which is a holomorphic map. If we write the map in coordinates therefore all $\tilde{\phi}_i$ are holomorphic functions.
Therefore we see near that $x$ on the neiborhood $\phi^{-1}(U_0)$ there exist $g_0 = 1, g_1 = \tilde{\phi_1}(x),....,g_n = \tilde{\phi_n(x)}$ which are all holomorphic and not identically zero, and represent $\phi$ on that neiborhood.