It is ok to write your name in red pen, but try not to write your name in red pen. It is more formal to use a black pen. Although the red pen plays an irreplaceable role and significance for some needs, it will only be used in some specific occasions, such as teachers marking test papers and doctors canceling medical orders.
The name of a calligraphy work cannot be written in red, but it can be replaced by a seal. The seal is red, which can be accepted by people and plays an embellishment role.
Why not try to write your name in red ink?
There are two general reasons. First, whenever someone was sentenced to death in ancient times, they would write their names with chicken blood. The names of other prisoners must not be red, otherwise this mistake will not be trivial. If everyone knows this rule, normal people's names will definitely feel uncomfortable when they are written in red.
The second reason is that emperors in many dynasties in history marked the throne with red pens, so it is naturally impossible for ordinary people to use the same color as the emperor. This is taboo, the emperor must be unique.