This is the 2515th most frequent Polish word.
mateuszem
"Mateuszem" is the instrumental case of the Polish male name "Mateusz," meaning "with Mateusz" or "by Mateusz."
The word 'Mateuszem' is the instrumental form of the name 'Mateusz,' used here to indicate the means or accompaniment in the sentence.
Spotkałem się wczoraj z Mateuszem.
I met up with Mateusz yesterday.
The word 'Mateuszem' is again in instrumental form, specifying accompaniment in the context of going on a trip.
W weekend wybieram się na wycieczkę z Mateuszem.
I am going on a trip with Mateusz this weekend.
In this sentence, 'Mateuszem' indicates the individual with whom the conversation took place, employing the instrumental case.