Preservers of numerical range of matrix skew products

Let \(\mathcal {M}_{n}\) ${\mathcal{M}}_n$ be the set of \(n\times n\) $n\times n$ complex matrices. Complete descriptions are given of the maps on \(\mathcal {M}_{n}\) ${\mathcal{M}}_n$ leaving invariant the numerical range of different kind of binary operations on matrices such as the skew product, the skew Lie product, the skew Jordan product and the skew semi-triple product, with no surjectivity assumption on them.