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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 )

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    You are asking for fields with characteristic 2.2012-12-03
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    @akkkk: Why did you delete your answer?2012-12-03
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    @wj32: I don't know, I wanted to add that the only field satisfying that was $F_2$, but then I realized that's not true so I figured I'm not an expert after all ;)2012-12-03

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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 $F_2$.

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    What does that mean?2012-12-03
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    @RobertS.Barnes: It means that $1 + 1 = 0$, where $1$ is the multiplicative identity in the field.2012-12-03
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    Or to put it another way, a field has characteristic 2 if and only if it has ${\bf Z}_2$ as a subfield.2012-12-03
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    The notation $F_2$ indicates a field of characteristic 2?2012-12-03
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    I'm sorry, no, $F_2$ is the field of order 2, so just what you call $\mathbf{Z}_2$.2012-12-03