Notation for the exclusive choice from the set I have the following formula: $$ \Omega \subset \mathbb{R}^d, $$ where $d=2$ or $d=3$. But I find this style of notation quite cumbersome. Is there any way, how to note, that $d$ equals to 2 or 3 exclusively and in more elegant way?

You could write $\Omega\subset\mathbb{R}^2\lor \Omega\subset\mathbb{R}^3$, or $\Omega\in\mathcal{P}\left(\mathbb{R}^2\right)\bigcup\mathcal{P}\left(\mathbb{R}^3\right)$. Here $\mathcal{P}\left(S\right)$ denotes the power set of $S$, i.e. its set of subsets. (Note $S\in\mathcal{P}\left(S\right)$. If your use of $\subset$ was meant to imply proper subsets, that second option should be amended to $\Omega\in\mathcal{P}\left(\mathbb{R}^2\right)\bigcup\mathcal{P}\left(\mathbb{R}^3\right)\backslash\left\\{\mathbb{R}^2,\,\mathbb{R}^3\right\\}$.)