I'll flesh out my hint. Suppose that $\alpha$ is a global holomorphic $p$-form on $\mathbb{P}^{n-1}$. Pullback $\alpha$ along the map $\mathbb{C}^{n} \setminus \{ 0 \} \to \mathbb{P}^{n-1}$; let the pulled back form be $\beta = \sum f_I(x_1, \ldots, x_n) d x_{i_1} \wedge \cdots d x_{i_p}$. The functions $f_I$ are holmorphic functions on $\mathbb{C}^{n} \setminus \{ 0 \}$; by Hartog's Lemma, they extend to holomorphic functions on $\mathbb{C}^n$.
The fact that $\beta$ is pulled back from $\mathbb{P}^{n-1}$ means that $\beta$ must be invariant under dilation. I.e. $\delta_t^{\ast} \beta = \beta$ where $\delta_t$ is the map $(x_1, \ldots, x_n) \mapsto (t x_1, \ldots, t x_n)$. Translating into specific equations, this shows that $f_I(t x_1, \ldots, t x_n) = t^{-p} f_I(x_1, \ldots, x_n)$.
If $f_I(x_1, \ldots, x_n)$ is nonzero, this implies that $\lim_{t \to 0} f_I(t x_1, \ldots, t x_n) = \infty$. This contradicts that $f_I$ extends to a homolorphic function at $0$.