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> いい **ん** ですか。
>
> いいですか。

Both sentences translate to "Is is OK?" ( _context_ : It is OK if I take this
umbrella?)

According to my textbook, んです is used for explanations. But in this context
someone is asking for permission, as far as I understand. Does ん make the
question sound more polite?
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question
grammar questions
んです for asking for permission
の/ん in a question sentence is for asking for a clarification. See: [Question
Markers: か and の](https://www.wasabi-jpn.com/japanese-grammar/question-
markers/)

> The connotation that the の has is to seek clarification based on background
> contexts, while the か doesn’t require any context.

> The plain question marker か allows you to just ask yes or no while the
> connotative question marker の allows you to clarify something that you doubt
> about. You may sometimes be rude if you suddenly try to clarify something
> without contexts.

When you have a previous context and are asking "is it (really) okay?"
referring to it, いいんですか is the natural choice. For example, when someone said
to you "Hey, you can take that umbrella!", then your response should be
いいんですか. But you have to say この傘を持って行っていいですか without の/ん if this is the first
time you mention 傘 in a conversation.
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answer