Suppose I have a discrete sequence (or would it be a set?) of numbers labeled by two integer indices, $\{a_{ik}|i,k\in 1,\ldots,n\}$. Also, I know that these numbers satisfy the properties
$a_{ij} + a_{jk} = a_{ik}$
and
$a_{ik} = -a_{ki}$
Are these conditions sufficient for me to conclude that there exists a sequence $\{\lambda_b|b\in 1,\ldots,n\}$ which satisfies
$a_{ik} = \lambda_i - \lambda_k$
for all $i,k$? If so, are both conditions necessary? If not, what other conditions would I require?
I really have no idea what tags to put on this, so I invite someone to retag it appropriately and remove this sentence.