  We develop a framework to compute characteristic classes and their forms in
the computer algebra system SageMath using symbolic calculus. In order to do
this, we make use of the Chern-Weil approach in which characteristic classes of
vector bundles in the de Rham cohomology are obtained by arbitrary connections.
Along the way, we implement the notion of vector bundles, their sections and
connections as well as mixed differential forms in SageMath. We conclude by
discussing some application examples exposed in Jupyter Notebook and eventually
address the issue of computational cost.
