Fields where $A^t=A$ and $A^t=-A$ Are there fields other $Z_2$ where there are matrices other than the zero matrix which are both symmetric and anti-symmetric at the same time? 
( $Z_2$ is {0,1} with modulo 2 addition and multiplication )

Any field with characteristic 2 has this property.
As you may recall, the characteristic of a ring $R$ is the smallest positive number $n$ such that $\sum_1^n 1_R=0$.
Concretely, let $A$ be a matrix over a field of characteristic 2, then $A^T$  also is, but $A^T+A^T=2A^T=0$ so $A^T$ is an additive inverse for $A^T$, which we also denote as $A^T=-A^T$.
Examples of other fields of characteristic 2 are: rational functions over $F_2$, the algebraic closure of $F_2$, and Laurent series over $F_2$.
