Concrete example of the Schur-Weyl duality for small representations

Question

I get that the Schur-Weyl duality states that

$$\mathbb{C}^2\otimes\mathbb{C}^2=S^2 \mathbb{C}^2 \oplus\Lambda^2\mathbb{C}^2$$

but what is a concrete example of decomposing a specific element, say $a\otimes b \in \mathbb{C}^2 \otimes \mathbb{C}^2$ as an element of $S^2 \mathbb{C}^2 \oplus\Lambda^2\mathbb{C}^2$?

Answer 1

$$a\otimes b=\underbrace{\tfrac12(a\otimes b+b\otimes a)}_{S^2\mathbb C^2}+\underbrace{\tfrac12(a\otimes b-b\otimes a)}_{\bigwedge^2\mathbb C^2}$$