So given are 3 vectors in $\mathbb{Z}_{3}^{3}$:
$$v_{1}=\begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}, v_{2}=\begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix}, v_{3}=\begin{pmatrix} 0\\ 1\\ -1 \end{pmatrix}$$
I first need to write them different because we are in $\mathbb{Z_{3}}$ where $-1$ is not allowed I think.
So we have the vectors:
$$v_{1}=\begin{pmatrix} 1\\ 1\\ 1 \end{pmatrix}, v_{2}=\begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix}, v_{3}=\begin{pmatrix} 0\\ 1\\ 2 \end{pmatrix}$$
Now is my actual question, am I allowed to write these 3 vectors like this?
$$\begin{pmatrix} 1 & 1 & 0\\ 1 & 0 & 1\\ 1 & 0 & 2 \end{pmatrix}$$
So is it allowed to take these 3 vectors and convert them to a $3\times3$ matrix?
I have looked this up on the internet and I only got more confused. So a yes or no would be enough for me on this question. And maybe you can also tell me if I changed the vectors correctly (-1 to 2), thank you very much :)
You have that $-1\equiv 2$ mod $3$, so yes, you may change it if you want. On the other hand, you can always write vectors as the columns of a matrix, but normally there is a reason to do it. For instance, if you wanted to see whether they are linearly independent or not (studying the rank of the matrix, for example). Or like Bernard said in the comments, you may take that as the change of basis matrix.