What is the matrix of $A$? We know the following about the linear map $A$: $\mathbb{R}$$^3$ -> $\mathbb{R}$$^3$:
$A$ is orthogonal 
$A$(1,2,2) = (1,2,2)
The vector (2,0,-1) is eigenvector for eigenvalue -1
dim $E_1$ = 1
Determine the matrix of $A$
I'm not quite sure which properties to use, such that i can create  a matrix $A$. Any help/tips? on proceeding this particular question?

From the given conditions you have two equations $Av_1=v_1$ and $Av_2=-v_2$.  
Notice that here additionally $v_1^Tv_2=0$ what means that both vectors are orthogonal.   
You can find also  transformed the third vector $v_3$ using as input vector cross product of $v_1$ and $v_2$, the result vector is orthogonal to both $= \pm (v_1 \times v_2)$ (transformation with orthogonal matrix preserves lengths and angles of vectors )
With this you have transformation
$A[v_1 \ \ v_2 \ \ v_1 \times v_2] = [v_1 \ \ -v_2 \ \ \pm v_1 \times v_2]$ what leads to direct calculation of    
$A= [v_1 \ \ -v_2 \ \ \pm v_1 \times v_2][v_1 \ \ v_2 \ \ v_1 \times v_2]^{-1} $ 
(two solutions).
Additionally if $\text{dim} \ E_1= 1$ means that $1$ is the eigenvalue with multiplicty $1$ then you can exclude  one solution. (the $-1$ is then eigenvalue with multiplicity $2$)
