Now that the question makes sense, let me mention the following result (you can find it in Eisenbud, Exercise 20.17) which hinted at what Francesco wrote.

Let $R = k[x,y,z]$ and $f\in m=(x,y,z)$. Then $f$ is a determinant of a matrix (of size at least $2$) with entries in $m$ iff $R/(f)$ is not a UFD!

Now that the question makes sense, let me mention the following result (you can find it in Eisenbud, Exercise 20.17) which hinted at what Francesco wrote.

Let $R = k[[x,y,z]]$ and $f\in m=(x,y,z)$. Then $f$ is a determinant of a matrix (of size at least $2$) with entries in $m$ iff $R/(f)$ is not a UFD!

Now that the question makes sense, let me mention the following result (you can find it in Eisenbud, Exercise 20.17) which hinted at what Francesco wrote.

Let $R = k[x,y,z]$ and $f\in m=(x,y,z)$. Then $f$ is a determinant of a matrix with entries in $m$ iff $R/(f)$ is not a UFD!

Now that the question makes sense, let me mention the following result (you can find it in Eisenbud, Exercise 20.17) which hinted at what Francesco wrote.

Let $R = k[x,y,z]$ and $f\in m=(x,y,z)$. Then $f$ is a determinant of a matrix (of size at least $2$) with entries in $m$ iff $R/(f)$ is not a UFD!