Maximum flow in a graph, and conservation of flow
The definition given in the lectures you've seen is the usual, which can be seen as the conservation of flow through a node, as it follows that flow exiting the node is equal to the flow entering the node. The expression doesn't only consider outgoing flow, as the equation $f(u,v) = -f(v,u)$ means that incoming flow can be described as 'negative' outgoing flow. 

Your expression currently just says 'the flow through $u$ is the flow through $u$', which doesn't restrict the flow at all.

A better version of the expression you try to pose is the following: 

$\begin{equation}
\sum_{v\in V_{in}(u)} f(v,u) = \sum_{v\in V_{out}(u)} f(u,v),
\end{equation}$

where $V_{in}(u)$ is the set of nodes $v$ that such that $v\rightarrow u$ is an edge and $V_{out}(u)$ the set of nodes $v$ such that $u\rightarrow v$ is an edge. 

It can be easily shown that this equation is equivalent to $\sum_{v\in V} f(u,v) = 0$ when $f(u,v)= -f(v,u)$ for all $u,v\in V$.