Broadly speaking, manipulation-related skill check shouldn't be used on players. One of the important tenets of D&D is the idea of player agency, where you always have control of what your character thinks, feels, and does. There are some exceptions to this rule (like mind-altering magic), but the only time your character should feel intimidated is when you, the player, judge them to be intimidated.

So, if another player was lying to you, you'd still do the normal opposed Deception/Insight. However, if a player was trying to convince you of something, he would have to actually convince you of that thing through roleplay, not just by making a Persuasion check.

Even if you like resolving things using skill checks most of the time, you shouldn't be able to control another player's actions through a skill check. Someone's roleplay shouldn't be dictated by the Rogue just because he has a +10 in Deception and Persuasion.

If I were the DM in that situation, I would ask the two of you to try working out the problem through roleplay without attacking each other, knowing that you are party members who trust each other with your lives on a regular basis. Be sure not to fall into the My Guy Syndrome trap where you tank the session because you feel like your character would never back down, but it's a lot more interesting to come to an actual conclusion rather than just having a skill check decide a social interaction between players.

I would suggest that you bring your problems to your DM, and talk to him about it before your next session, so that you can have a better agreement between the two of you about how to deal with similar problems in the future.

Broadly speaking, manipulation-related skill check shouldn't be used on players. One of the important tenets of D&D is the idea of player agency, where you always have control of what your character thinks, feels, and does. There are some exceptions to this rule (like mind-altering magic), but the only time your character should feel intimidated is when you, the player, judge them to be intimidated.

So, if another player was lying to you, you'd still do the normal opposed Deception/Insight. However, if a player was trying to convince you of something, he would have to actually convince you of that thing through roleplay, not just by making a Persuasion check.

Even if you like resolving things using skill checks most of the time, you shouldn't be able to control another player's actions through a skill check. Someone's roleplay shouldn't be dictated by the Rogue just because he has a +10 in Deception and Persuasion.

This isn't to say that you can't allow other party members to intimidate your character with a roll, if that's the kind of dynamic that your group is comfortable with, but you shouldn't have your agency taken away without knowing up-front that that's something you can expect. There is no such thing as badwrongfun, but anything that involves loss of agency needs a considered approach.

If I were the DM in that situation, I would ask the two of you to try working out the problem through roleplay without attacking each other, knowing that you are party members who trust each other with your lives on a regular basis. Be sure not to fall into the My Guy Syndrome trap where you tank the session because you feel like your character would never back down, but it's a lot more interesting to come to an actual conclusion rather than just having a skill check decide a social interaction between players.

I would suggest that you bring your problems to your DM, and talk to him about it before your next session, so that you can have a better agreement between the two of you about how to deal with similar problems in the future.