  Vector algebra is a powerful and needful tool for Physics but unfortunately,
due to lack of mathematical skills, it becomes misleading for first
undergraduate courses of science and engineering studies. Standard vector
identities are usually proved using Cartesian components or geometrical
arguments, accordingly. Instead, this work presents a new teaching strategy in
order to derive symbolically vector identities without analytical expansions in
components, either explicitly or using indicial notation. This strategy is
mainly based on the correspondence between three-dimensional vectors and
skew-symmetric second-rank tensors. Hence, the derivations are performed from
skew tensors and dyadic products, rather than cross products. Some examples of
skew-symmetric tensors in Physics are illustrated.
