Almost nobody does this without a reason, and almost everybody underestimates how it will look to their manager if it's done the wrong way. The reason matters less than the method.
There are legitimate reasons to bring something directly to someone above your manager: a genuine ethics or safety concern your manager isn't addressing, a decision that's explicitly above your manager's authority and needs a different owner, or a situation where you've raised something with your manager repeatedly and nothing has changed. None of these require going around your manager secretly, the method matters as much as the reason.
Going around your manager rarely damages the relationship. Going around them without telling them almost always does.
The version that preserves trust, even when the topic is uncomfortable, is telling your manager directly that you plan to raise something above them, and ideally why, before you do it, not asking permission, necessarily, but not doing it silently either. Most managers, even ones who'd rather you didn't escalate past them, respect being told directly far more than finding out after the fact that they were bypassed.
"I want to be straightforward with you, I'm planning to raise the budget concern directly with Sarah, since it's ultimately her call and I don't think it's something you can resolve on your own. Wanted you to hear that from me first, not find out after."
That message doesn't ask permission, and it doesn't need to, but it removes the element of surprise, which is usually what actually damages the relationship, more than the act of escalating itself. A manager who's told in advance can prepare, respond, or even offer to make the introduction themselves. A manager who finds out secondhand experiences it as a betrayal regardless of how legitimate the underlying reason was.
The exception worth naming clearly: if the concern is about your manager's own conduct specifically, something unethical, unsafe, or a pattern of behavior you can't raise with them directly, the calculus changes, and going to HR or their own manager without a heads-up is reasonable, sometimes necessary. That's a different situation from a disagreement about strategy or a budget call, where transparency about your intent to escalate is almost always the better path.