Why Set theory without the axiom of foundation? is an article explaining an alternative implementation of ordered pairs, apparently due to Quine. It's much less obvious than the Kuratowski definition, but has the added benefit that $x = \langle z, x \rangle$ can be true without contradicting the standard Axiom of Foundation (the linked paper goes into more detail as to why and how you'd do that).
To describe the construction, I'll need some notation: $s$ is the function that takes a finite von Neumann ordinal (i.e. a member of $\omega$) and adds 1 to it, leaving everything else untouched, $z(x) = x \cup \{0\}$, and if $f$ is a function then $f“x$ is the set $\{ f(y) : y \in x \}$.
Then $s“x$ and $s“y$ are sets from which we can easily recover $x$ and $y$, and which don't contain $0$.
Now, the punchline: $\langle x, y \rangle = s“x \cup z“(s“y)$.
$s“x$ is recoverable as $\{z \in \langle x, y \rangle : 0 \not\in z\}$;
$s“y$ is $\{z \setminus \{0 \}: z \in \langle x, y \rangle, 0 \in z \}$.
This defines projections $\pi_1$ and $\pi_2$ onto the first and second components respectively, so $\langle a,b \rangle = \langle c,d \rangle \implies a = c \wedge b = d$. You can check that $x \mapsto \langle \pi_1(x), \pi_2(x) \rangle$ is the identity, so that the converse holds.