This paper has aim to characterize Fredholmness and Weylness of upper triangular operator matrices having arbitrary dimension n ≥ 2. We present various characterization results in the setting of infinite dimensional Hilbert spaces, thus extending some known results from Cao X. et al. (Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 1, 169–178 and J. Math. Anal. Appl. 304 (2005), no. 2, 759–771) and Zhang et al. (J. Math. Anal. Appl. 392 (2012), no. 2, 103–110) to the case of arbitrary dimension n ≥ 2. We pose our results without using separability assumption, thus improving perturbation results from Wu X. et al. (Ann. Funct. Anal. 11 (2020), no. 3, 780–798 and Acta Math. Sin. (Engl. Ser.) 36 (2020), no. 7, 783–796).