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For what follows, I recommend SGA 2 (available on Arxiv). There is an exact sequence $$0\rightarrow H^1(X,\mathcal{O}_X)\rightarrow H^1(X\smallsetminus p,\mathcal{O}_X)\rightarrow H^2_{p}(X,\mathcal{O}_X)\rightarrow H^2(X,\mathcal{O}_X)$$ where $ H^2_{p}(X,\mathcal{O}_X)$ is infinite-dimensional, while $H^i(X,\mathcal{O}_X)$ is finite-dimensional. Therefore $H^1(X\smallsetminus p,\mathcal{O}_X)$ is infinite-dimensional. The computation of $ H^2_{p}(X,\mathcal{O}_X)$ given in SGA 2 and the exact sequence above give an expression for this space, probably not very pleasant.