For $A \in SU(2,C)$, it is clear that $A$ is completly determined by its first row (well any row or column, but let's say first column). In the general $SU(n,C)$-case this is no longer true. In fact, it seems that for every complex $n$-vector of unit norm, there exists a family of matrices for which the vector is the first column. By working with messy simultaneous equations I seem to have shown that every two elements of this family differ by a multiple of $SU(n-1)$ embedded in the bottom right hand corner with 1 in the first entry of the first column and zeros everywhere else. I suspect that there is a bijective correspondence between elements of this family and $SU(n-1)$ but can't prove it.

Can anyone give a simple slick reworking of all of this?