Superconformal algebras are superextensions of the Virasoro algebra. They play an important role in the string theory and conformal field theory. Central extensions of contact superalgebras $K(2)$ and $K'(4)$ are known to physicists as the $N = 2$ and the big $N = 4$ superconformal algebras. A remarkable property of $\widehat{K}'(4)$ and the exceptional superconformal algebra $CK_6$ is that they admit embeddings into the Lie superalgebras of pseudodifferential symbols on the circle, extended by $N = 2, 3$ odd variables. Associated to these embeddings, there are ``small" irreducible representations of these superalgebras and their realizations in matrices of size $2^N$ over a Weyl algebra. The general construction of such matrix realizations is connected with the spin representation of $\mathfrak{o}(2N + 1, \mathbb{C})$. We also obtain a realization of the family of simple exceptional finite-dimensional Lie superalgebras $D(2; 1; \alpha)$, related to $K(4)$ in matrices over a Weyl algebra.