Is there a straightforward generalization of the $\mathbb{C}^2$ Bell basis to $N$ dimensions? Is there a rotational invariant Bell state in higher dimensions? If yes, then what is the form of that state (how does it look like)? And, by rotational invariance, I mean that the state is invariant under applying the same unitary transformation $U$ to each qubit separately.
For example, $|\psi^-\rangle = \frac{|0\rangle|1\rangle - |1\rangle|0\rangle}{\sqrt{2}} = \frac{|\gamma\rangle|\gamma_\perp\rangle - |\gamma_\perp\rangle|\gamma\rangle}{\sqrt{2}}$, where $|\gamma\rangle$ is some quantum state in $\mathbb{C}^2$, and $|\gamma_\perp\rangle$ is orthogonal to $|\gamma\rangle$.
It would be helpful if I could see an example of the same, in say $\mathbb{C}^4$ space, perhaps in the computational basis {$|0\rangle, |1\rangle, |2\rangle, |3\rangle$} itself.