I saw written in a coat of arms "PRO MARE NOSTRVM", but we all know that the preposition "pro" takes ablative, so the right form would be "PRO MARI NOSTRO" wouldn't it?
I could not find anywere other uses for the preposition "pro". Is that sentence somehow right?
The entry for pro in Lewis & Short mentions at II that the preposition pro comes with the ablative but remarks that accusative is possible in late Latin. As you quote a coat of arms, influences of late Latin are certainly a possibility. I don't know what the relative frequency of the two cases with pro is in any given era — apart from the accusative having zero frequency in classical Latin — but I would say that what you found is not a singular outlier or a mistake but rather a feature of later Latin.
The ablative ending of the third declension is -e or -i, and some words admit both endings. As rjpond mentions in a comment, mare is an attested ablative form. However, pro mare nostrum has an unambiguous accusative because of the nostrum. I would regard mare nostro as a classical alternative ablative, but mare nostrum is an accusative.
Another possible interpretation would be that pro (alternatively written proh) is an interjection of sorrow or desperation, meaning something like “alas,” “alack”. While often used with a nominative or vocative (pro dii immortales, pro sancte Juppiter etc), it is also found with the accusative: pro deorum hominumque fidem! (which makes sense because the accusative is generally often used for exclamations).
Therefore, pro mare nostrum can be interpreted as:
Alas, our sea!
Whether that is a sensible reading in a coat of arms is debatable.
This means "Ours by the sea"* (or, if you want to supply a noun where Latin will let an adjective do the job of a noun "Our place by the sea"). Thus, nostrum is nominative and mare ablative. It would be a nice motto for a family with a sea-side home, or for a sea-side city.
Alternative interpretations, with nostrum modifying mare, are way too grandiose for most coats of arms. To call a body of water mare nostrum, you need to surround it entirely. Turkey might refer thus to the Sea of Marmara, but this is the motto of a Brazilian city on the Atlantic, which pretty well excludes that meaning.
Nonetheless, most places on the internet (including Wikipedia) translate this motto as "for our sea" or the like, which just shows that you shouldn't rely on the internet to interpret Latin. People who don't know Latin are probably relying on the familiarity of the phrase mare nostrum from Mussolini's use of it to refer (very) aspirationally to the Mediterranean. The fact that the syntax with pro doesn't work out that way should in this case be an additional clue that you need to reconstrue.
For mare as an ablative, see A&G 76a3.
*Literally "in front of the sea" or "before the sea" but English idiom in this circumstance prefers "by the sea".
