  For a skew left brace $(G, \cdot, \circ)$, the map $\lambda : (G, \circ) \to
\Aut \,(G, \cdot),~~a \mapsto \lambda_a,$ where $\lambda_a(b) = a^{-1} \cdot (a
\circ b)$ for all $a, b \in G$, is a group homomorphism. Then $\lambda$ can
also be viewed as a map from $(G, \cdot)$ to $\Aut \, (G, \cdot)$, which, in
general, may not be a homomorphism. A skew left brace will be called
$\lambda$-anti-homomorphic ($\lambda$-homomorphic) if $\lambda : (G, \cdot) \to
\Aut \, (G, \cdot)$ is an anti-homomorphism (a homomorphism). We mainly study
such skew left braces. We device a method for constructing a class of binary
operations on a given set so that the set with any two such operations
constitute a $\lambda$-homomorphic symmetric skew brace. Most of the
constructions of symmetric skew braces dealt with in the literature fall in the
framework of our construction. We then carry out various such constructions on
specific infinite sets.
