Planen er skør.

Because planen is the definite singular form of plan.

• en plan = a plan
• planen = the plan

In Danish, the definite article is often attached to the end of the noun instead of being a separate word.

So:

• Planen er skør = The plan is crazy

If you said Plan er skør, that would be ungrammatical in normal Danish.

Danish often expresses the by adding an ending to the noun.

For a common-gender noun like plan, the definite ending is usually -en:

• en plan → planen

A separate definite word such as den is usually used when there is an adjective before the noun:

• den skøre plan = the crazy plan

But when the noun stands alone, Danish normally uses just the ending:

• planen

Er is the present tense of at være, which means to be.

So:

• jeg er = I am
• du er = you are
• planen er = the plan is

In this sentence, er works exactly like English is.

Because Danish adjectives often agree with the noun in gender and number.

Plan is a common-gender noun, so the predicate adjective takes the common-gender singular form:

• en plan er skør

If the noun were neuter, you would usually add -t:

• et forslag er skørt = a proposal is crazy

And in the plural, you usually get -e:

• planer er skøre = plans are crazy

So in Planen er skør, skør is the correct form because planen is singular and common gender.

No. This is a very basic Danish sentence pattern:

subject + verb + adjective/complement

So:

• Planen = subject
• er = verb
• skør = complement

This is very similar to English:

• The plan is crazy

Danish does have a verb-second rule in main clauses, but this sentence already follows that naturally because the subject comes first and the verb comes second.

Skør can be tricky because of ø, which does not exist in English.

A rough guide:

• sk- sounds like sk in skill
• ø is a rounded front vowel; there is no exact English equivalent
• the r affects the vowel and may be quite soft depending on accent

A very rough approximation for an English speaker is something like skurh or skerh, but neither is exact.

The most important part is the ø:

• keep your tongue somewhat like for eh
• round your lips

If you are learning standard Danish pronunciation, listening and repeating is especially important for words with ø.

It can mean several things depending on context.

With something like a plan, skør usually means:

• crazy
• wild
• ridiculous
• nuts in an informal sense

It does not usually suggest a clinical or medical meaning here. In Planen er skør, it normally means the plan seems absurd, unrealistic, or strangely bold.

So the tone is often about the idea, not about mental illness.

Yes, but it means something different.

• Planen er skør = The plan is crazy
  This refers to one specific plan.

• En plan er skør = A plan is crazy
  This sounds less natural by itself unless the context already makes it clear which one you mean, or unless you are contrasting it with another plan.

So if you are talking about a particular plan already known in the conversation, planen is the natural choice.

Plan is a common-gender noun.

That means its indefinite article is en, not et:

• en plan
• planen

This matters because gender affects:

• the article
• the definite ending
• adjective agreement in some contexts

For example:

• en skør plan = a crazy plan
• planen er skør = the plan is crazy

If plan were a neuter noun, you would expect et and often adjective forms like skørt, but that is not the case here.