  In this paper, we explore linear representations of skew left braces, which
are known to provide bijective non-degenerate set-theoretical solutions to the
Yang--Baxter equation that are not necessarily involutive. A skew left brace
$(A, \cdot, \circ)$ induces an action $\lambda^{\op}: (A, \circ) \to \Aut (A,
\cdot)$, which gives rise to the group $\Lambda_{A^{\op}} = (A, \cdot)
\rtimes_{\lambda^{\op}} (A, \circ)$. We prove that if $A$ and $B$ are isoclinic
skew left braces, then $\Lambda_{A^{\op}}$ and $\Lambda_{B^{\op}}$ are also
isoclinic under some mild restrictions on the centers of the respective groups.
Our key observation is that there is a one-to-one correspondence between the
set of equivalence classes of irreducible representations of $(A, \cdot,
\circ)$ and that of the group $\Lambda_{A^{\op}}$. We obtain a decomposition of
the induced representation of the additive group $(A, \cdot)$ and of the
multiplicative group $(A, \circ)$ corresponding to the regular representation
of the group $\Lambda_{A^{\op}}$. As examples, we compute the dimensions of the
irreducible representations for several skew left braces with prime power
orders.
