We use Chern–Weil theory for Hermitian holomorphic vector bundles with canonical connections for explicit computation of the Chern forms of trivial bundles with special non-diagonal Hermitian metrics. We prove that every $$\bar{\partial }\partial $$ -exact real form of the type $$(k,k)$$ on an $$n$$ -dimensional complex manifold $$X$$ arises as a difference of the Chern character forms of trivial Hermitian vector bundles with canonical connections, and that modulo $$\mathrm {Im}\,\partial +\mathrm {Im}\,\bar{\partial }$$ every real form of type $$(k,k)$$ , $$k<n$$ , arises as a Bott–Chern form for two Hermitian metrics on some trivial vector bundle over $$X$$ . The latter result is a complex manifold analogue of Proposition 2.6 in the paper by Simons and Sullivan (Am Math Soc 11:579–599, 2010). As an application, we obtain an explicit formula for the Bott–Chern form of a short exact sequence of holomorphic vector bundles considered by Bott and Chern (Acta Math 114:71–112, 1965), for the case when the first term is a line bundle.