Use RSA private key to generate public key?

openssl genrsa -out mykey.pem 1024

will actually produce a public - private key pair. The pair is stored in the generated mykey.pem file.

openssl rsa -in mykey.pem -pubout > mykey.pub

will extract the public key and print that out. Here is a link to a page that describes this better.

EDIT: Check the examples section here. To just output the public part of a private key:

openssl rsa -in key.pem -pubout -out pubkey.pem

To get a usable public key for SSH purposes, use ssh-keygen:

ssh-keygen -y -f key.pem > key.pub

In most software that generates RSA private keys, including OpenSSL's, the private key is represented as a PKCS#1 RSAPrivatekey object or some variant thereof:

A.1.2 RSA private key syntax

An RSA private key should be represented with the ASN.1 type
RSAPrivateKey:

  RSAPrivateKey ::= SEQUENCE {
      version           Version,
      modulus           INTEGER,  -- n
      publicExponent    INTEGER,  -- e
      privateExponent   INTEGER,  -- d
      prime1            INTEGER,  -- p
      prime2            INTEGER,  -- q
      exponent1         INTEGER,  -- d mod (p-1)
      exponent2         INTEGER,  -- d mod (q-1)
      coefficient       INTEGER,  -- (inverse of q) mod p
      otherPrimeInfos   OtherPrimeInfos OPTIONAL
  }

As you can see, this format has a number of fields including the modulus and public exponent and thus is a strict superset of the information in an RSA public key.

People looking for SSH public key...

If you're looking to extract the public key for use with OpenSSH, you will need to get the public key a bit differently

$ ssh-keygen -y -f mykey.pem > mykey.pub

This public key format is compatible with OpenSSH. Append the public key to remote:~/.ssh/authorized_keys and you'll be good to go

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docs from SSH-KEYGEN(1)

ssh-keygen -y [-f input_keyfile]  

-y This option will read a private OpenSSH format file and print an OpenSSH public key to stdout.