How to create an oracle dump(.dmp) file using Toad?
What is your Toad version?

I have 9.5 here and there are three options:


Database > Export > Export Utility Wizard
Database > Export > Export DDL
Database > Export > Generate Database Script


There's one last option, in case you only need to get the 'skeleton' for one schema:


Database > Export > Generate schema script


Have you tried any of that?

[EDIT]
I've tried Alex Poole approach and it didn't work for me either.

Sqlplus to the entry:

p08[oracle] (sigepsti) /oracle/product/11.2.0/db/network/admin$ sqlplus system@sigepsti

SQL*Plus: Release 11.2.0.4.0 Production on Thu Jul 31 17:07:23 2014

Copyright (c) 1982, 2013, Oracle.  All rights reserved.

Enter password:

Connected to:
Oracle Database 11g Enterprise Edition Release 11.2.0.4.0 - 64bit Production
With the Partitioning, Automatic Storage Management, OLAP, Data Mining
and Real Application Testing options

(sigepsti)SQL&gt; exit


Expdp error (exp gets the same error):

p08[oracle] (sigepsti) /oracle/product/11.2.0/db/network/admin$ tnsping sigepsti

TNS Ping Utility for Linux: Version 11.2.0.4.0 - Production on 31-JUL-2014 17:07:37

Copyright (c) 1997, 2013, Oracle.  All rights reserved.

Used parameter files:
/grid/product/11.2.0/grid/network/admin/sqlnet.ora


Used TNSNAMES adapter to resolve the alias
Attempting to contact (DESCRIPTION = (ADDRESS = (PROTOCOL = TCP)(HOST = dborasigepsti)(PORT = 1521)) (CONNECT_DATA = (SERVER = DEDICATED) (SERVICE_NAME = sigepsti)))
OK (0 msec)
p08[oracle] (sigepsti) /oracle/product/11.2.0/db/network/admin$ expdp system@'dborasigepsti:1521/sigepsti'

Export: Release 11.2.0.4.0 - Production on Thu Jul 31 17:07:50 2014

Copyright (c) 1982, 2011, Oracle and/or its affiliates.  All rights reserved.

UDE-12504: operation generated ORACLE error 12504
ORA-12504: TNS:listener was not given the SERVICE_NAME in CONNECT_DATA