How can I find a resolution of $\Omega^i_X$ for a projective smooth complete intersection $X$?

Question

Given a projective smooth complete intersection $i:X \to \mathbb{P}^n$ defined by polynomials $(\underline{f}) = (f_1,\ldots,f_k)$ the koszul complex $K^\bullet(\underline{f})$ can be used to compute the sheaf cohomology $H^k(X,\mathcal{O}_X)$. This is because $$ H^k(X,\mathcal{O}_X) = H^k(\mathbb{P}^n,i_*\mathcal{O}_X) $$ and the second term can be computed using the spectral sequence $$ E_1^{p,q} = H^q(\mathbb{P}^n,K^{p}) $$ Does there exist a "nice" flat resolution of $\Omega_X^k$ which gives a similar type of spectral sequence?